một số bài toán ngược cho phương trình elliptic với quan sát biên some inverse problems for elliptic equations with boundary observations

Rn n− dimensional Euclidean space; Ω open set in Rn; ∂Ω boundary of Ω; Γ subset of ∂Ω; Q = Ω × 0, T S = ∂Ω × 0, T; CQ space of continuous functions on Q; CkQ space of k-times continuousl

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

LE THI THU GIANG

MỘT SỐ BÀI TOÁN NGƯỢC CHO PHƯƠNG

TRÌNH ELLIPTIC VỚI QUAN SÁT BIÊN (SOME INVERSE PROBLEMS

FOR ELLIPTIC EQUATIONS WITH BOUNDARY OBSERVATIONS)

THESIS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS HA NOI –

2024

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

LE THI THU GIANG

MỘT SỐ BÀI TOÁN NGƯỢC CHO PHƯƠNG TRÌNH ELLIPTIC VỚI QUAN SÁT BIÊN

(SOME INVERSE PROBLEMS

FOR ELLIPTIC EQUATIONS WITH

DINH NHO HAO

HA NOI – 2024

In Chapter 2, we study the source identification problem for elliptic equations with observation on the boundary We formulate the inverse problem as an operator equation andregularize it by the Tikhonov regularization method To discretize the problem, we followHinze’s idea on variational discretization and suggest a rule of choosing the regularizationparameter depending on the noise level in the observation data and the discretization meshsize which yields the optimal convergence rate This abstract result is applied to the finiteelement method for numerical solving the source identification problem Some numericalexamples are presented for showing the efficiency of the method

Chapter 3 is devoted to the problem of determining a term in the right-hand side ofelliptic equation with constant and variable coefficients in a cylinder from boundaryobservations Based on the special form of the considered equation in a cylinder, thesolution of the direct and inverse problems can be represented by the Fourier series Sincethe source problem is ill-posed, we regularize it using the truncated Fourier series methodand propose a method for selecting the number of Fourier coefficients to ensure theconvergence of the method and to indicate the rate of convergence To demonstrate theefficiency of the method, we present several numerical examples

i

Trong Chương 2, chúng tôi nghiên cứu bài toán xác định nguồn cho các phươngtrình elliptic với quan sát trên biên Bài toán này có thể viết dưới dạng phương trìnhtoán tử Chúng tôi đã sử dụng phương pháp hiệu chỉnh Tikhonov để chỉnh hóaphương trình toán tử và rời rạc nó dựa trên ý tưởng rời rạc biến phân (variationaldiscretization) của Hinze, rồi sau đó đề xuất một quy tắc chọn tham số hiệu chỉnhphụ thuộc vào mức độ nhiễu trong dữ liệu quan sát và bước lưới để nhận được tốc

độ hội tụ tối ưu Kết quả đã này được áp dụng cho phương pháp phần tử hữu hạn

để giải số bài toán xác định vế phải và được thực hiện bằng số trên máy tính

Trong Chương 3, chúng tôi xét bài toán xác định vế phải của một phương trìnheliptic với hệ số hằng và hệ số biến thiên trong hình trụ từ các quan sát trên biên.Dựa vào cấu trúc đặc biệt của phương trình đang xét và miền hình trụ, lời giải củabài toán thuận và bài toán ngược có thể biểu diễn qua chuỗi Fourier Do bài toán đặtkhông chỉnh nên chúng tôi hiệu chỉnh nó bằng phương pháp chặt cụt chuỗi Fourier

và đưa ra cách chọn số các hệ số Fourier để phương pháp hội tụ và chỉ ra tốc độhội tụ

Cuối mỗi chương, chúng tôi trình bày một số ví dụ số để thể hiện tính hiệu quả của các phương pháp đã đề xuất

ii

Declaration

This work has been completed at Institute of Mathematics, Vietnam Academy ofScience and Technology under the supervision of Prof Dr Habil Đinh Nho Hào Ideclare hereby that the results presented in it are new and have never beenpublished elsewhere

Author: Lê Thị Thu Giang

iii

Acknowledgments

First and foremost, I want to express my deepest gratitude to my advisor, Professor Đinh Nho Hào for his invaluable help and support in my research He spent alot of time and energy on my work as well as giving me many valuable opinions andcomments, and he gradually guided me to get acquainted with my scientific researchwork For me, he is not only an extremely respected teacher but also my second father, who always gives me a lot of support and patience, encourages me toovercome difficulties in my study and my daily life

I would like to express my special appreciation to Professor Hoàng Thế Tuấn, DrĐào Quang Khải, Dr Lương Thái Hưng and other members of the weekly seminar atDepartment of Differential Equations for many interesting discussions Also, I wouldlike to thank Professor Nguyễn Văn Đức (Vinh University), Professor Nguyễn TrungThành (Rowan University, USA) and all of friends in Professor Đinh Nho Hào’s groupseminar for their valuable comments and suggestions to my research papers

Thanks go to my sincere friend, Dr Nguyễn Thị Ngọc Oanh (College of Sciences,Thai Nguyen University) for her listening, offering me advice, and supporting methrough my entire study process I am very grateful to Dr Phan Xuân Thành (HanoiUniversity of Science and Technology) for his kind help on computer programmingand giving useful comments for improving my PhD thesis

My great appreciation is also expressed to the leaders of the Institute of Mathematics, the Center of Postgraduate Training and the International Center ofResearch and Postgraduate Training in Mathematics for providing me with such anexcellent study environment as well as financial support during my PhD study

I would like to thank the leaders of Thuong mai University, the Dean, as well asall of my colleagues at the Faculty of Mathematical Economics and my friends fortheir encouragement and support throughout my PhD study

Last but not least, this journey would not have been possible without the support

of my family I would like to express my sincere gratitude to my parents, my parentsin-law, my husband, my children, my brothers and sisters for their unconditional loveand encouragement to me Especially, this thesis is dedicated to my beloved fatherwho was my first teacher inspiring me to study Mathematics Although he passedaway, his boundless love, trust and hope for me are my motivation to complete thiswork

List of Figures

1.1 Example 1.5.1 Exact solution 33 1.2 Example 1.5.1 Numerical solution with p=0.01, a priori method 33 1.3 Example 1.5.1 Approximation error with p=0.01, a priori method 34 1.4 Example 1.5.1 Approximation error with p=0.035, a priori method 34

1.5 Example 1.5.1 Comparison of the exact solution with numerical solutions for various p, a priori method 34

1.6 Example 1.5.1 Comparison of the approximation errors of the numerical so lutions for various p, a priori method 34

1.7 Example 1.5.1 Exact and approximate solutions at T = 1 for various p, a posteriori

1.12 Example 1.5.3 Approximation errors at T = 1 for various p with x ∈ [0, π] 37

1.13 Example 1.5.3 Exact and approximate solutions at T = 1 for various p, x ∈ [0, 5π] 37

1.14 Example 1.5.3 Approximation errors at T = 1 for various p, x ∈ [0, 5π] 37 v

2.1 Example 2.4.1 Reconstruction of the smooth source function f1(x1) 65 2.2 Example 2.4.1 Reconstruction of the continuous and nonsmooth source func tion

f2(x1) 65 2.3 Example 2.4.1 Reconstruction

of the discontinuous source function f3(x1) 66 2.4 Example 2.4.1 Exact and

approximation solutions with f ∗ = 0, 10 and 20 67 2.5 Example 2.4.2 f 20x1+x2 and itsnumerical solution with f ∗close to f 20x1+x2 69 2.6 Example 2.4.2 f0and the numericalsolution with f ∗close to f0 70 2.7 Example 2.4.2 f −5 sin πx2 and the numericalsolution with f ∗close to f −5 sin πx2 71 2.8 Example 2.4.3 Numerical results for f satisfying the source condition (2.29) with θ =

parameters 66 2.4 Example 2.4.1 L2-error between the exact solution and numerical ones with relative noise 0.1%, 1% for f ∗ = 0, 10, and 20 67

2.5 Example 2.4.2 L2-errors with f ∗close to exact solutions 68 2.6 Example2.4.3 L2-error behavior - the source condition 1 72 2.7 Example 2.4.4 L2-error behavior - the source condition 2 75

3.1 Example 3.3.1 The L2-norm of relative errors for smooth function 90 3.2Example 3.3.2 The L2-norm of relative errors - non-smooth, continuous func tion 92 3.3 Example 3.3.3 The L2-norm of relative

errors for the discontinuous function 93

viii

List of Notations

Rn n− dimensional Euclidean space;

C(Q) space of continuous functions on Q;

C k (Q) space of k-times continuously differentiable functions on Q; C ∞

0(Q) space of infinitely differentiable functions

with compact support in Ω;

L2(Ω), L2(S), L2(Q) space of measurable, square-integrable functions in Ω (resp.

Abstract i Acknowledgments iv List of Notations ix Introduction 1

1 The Cauchy problem for elliptic equations 11 1.1 Problem settings and the non-local boundary problem regularization method 11 1.2 The very weak solution to the Cauchy problem for elliptic equations 14 1.3 The well-posedness of the non-local boundary

value problem for elliptic equations 18

1.3.1 The very weak solution 18 1.3.2 The classical

solution 20

1.4 Finite difference scheme for the non-local

boundary value problem 23 1.4.1 The three-dimensional problem 24 1.4.2 The two-dimensional

2.3 The inverse source problem and its finite element approximation 53 2.3.1The inverse problem 53 2.3.2 Conjugate GradientMethod 55 2.3.3 Finite element method for the directproblem 57 2.3.4 Regularized discretized variational problem

62 2.4 Numerical examples 62

3 Determining a source term in an elliptic equation in a cylinder from boundary

observations 76

3.1 Problem setting 76 3.1.1 A representation

of the solution 79 3.1.2 Regularization by the truncatedFourier series 81

3.2 A special case of equations with constant coefficients in a parallelepiped 83 3.2.1Problem setting 83 3.2.2 A representation to thesolution 84 3.2.3 Regularization by the truncated Fourier

series 87 3.3 Numerical examples

a direct problem includes

• the domain in which the process is studied;

• the equation (including the parameters and the relations) that describes the process; •the initial conditions (if the process is nonstationary);

• the conditions on the boundary of the domain

If all the inputs, parameters and relations between them are known in a process, and wewant to know how the process look like, we have a direct problem The difficulties of solvingthese problems cause by their ill-posedness Following Hadamard, [35], a problem is well-posed if

• there exists a solution,

• the solution is unique,

• the solution continuously depends on the data in an appropriate topology

Once, at least one of these conditions is violated we say that the problem is ill-posed In

1902 Hadamard [34] considered the Cauchy problem for the Laplace equation

sin(nx), n ∈ N ∗, the solution to (0.1) has the form u n(x, y) = e √nsin(nx) sinh(ny)

It is clear that, for a = 0, b = π, although ∥φ n ∥ L ∞ (0,π) , ∥ψ n ∥ L ∞ (0,π) → 0, ∥u n(., y)∥ L ∞ (0,π) → ∞,

as y > 0, n → ∞ Thus, the solution does not continuously depend on the Cauchy data Hence, the problem is ill-posed

At that time, Hadamard thought that ill-posed problems have no physical meaning However, many problems in science, technology and practice lead to ill-posed problems (see[32, 34, 54, 78, 94]) To our best knowledge, A N Tikhonov, [90, 109, 110], M M Lauvrent’ev, [75, 77–79], F John [59], C Pucci [99], V K Ivanov [55, 56] are the first whostudied ill-posed problems From their studies, one can see that while the first two conditions of the well-posedness conditions can be overcome by imposing an additional data orrestricting the set of admissible solutions, the third condition is very hard to overcome Sincethere always contain random noise in the data generated during the measurement process,the proposed problem cannot be satisfied exactly Even when the noise level in a measuredoutput data is small, many algorithms developed for well-posed problems do not work forunstable problems because of round-off errors Due to the instability of the problems, asmall perturbation in the data can cause very large errors in the numerical solutions.Therefore, if the stability condition does not hold, it is impossible to directly apply traditionalcompu tational methods for ill-posed problems This is the reason why regularizationmethods play a central role in the theory and applications of inverse and ill-posed problems.The purpose of regularization methods is to approximate the ill-posed problems by a series

of well-posed problems such that the solutions of these problems converge to the exactsolution

Two popular regularization methods are proposed by Lavrent’ev [76] and Tikhonov[109] A typical inverse problem can be written as the operator equation Af = g, where f isthe unknown which we have to seek, g is the data which is perturbed by noise (so, inpractice, this datum is replaced by g ϵ) and A is a compact operator mapping from a Hilbert

space X to a Hilbert space Y The idea of Lavrent’ev was that, instead of taking f ϵ =

A −1 g ϵas a solution, he approximated it by (A + αI)−1 g ϵ with α = α(ϵ) > 0 being aregularization

2parameter such that when ϵ → 0, then α → 0 and f ϵ → f

In 1963, Tikhonov [109] published the well-known Tikhonov regularization method by minimizing the following functional with respect to f

∥Af − g ϵ ∥2Y + αl(f),

where l(f) ≥ 0 is a bounded regularization functional It is proved that this optimizationproblem is well-posed and its solution f ϵtends to the exact solution f when ϵ → 0 if α isproperly chosen Besides, by using stability estimates, the convergence rate can beobtained Since then, the theory of ill-posed problems had been extensively developed [32,

54, 78, 94] Moreover, ill-posed and inverse problems have become an independent branch

• The regularization, least squares methods [81, 88, 109, 112, 119, 120]

Among inverse problems in PDEs, those for elliptic equations are of great interest to manyresearchers because of their crucial role in practice These problems have been studied fordecades, yielding a wealth of interesting results in both quantitative and qualitative aspects.For surveys on inverse problems for elliptic equations, we refer to [17, 21, 29, 67, 106, 107]for identifying coefficients, [49] for identifying boundary values, [62, 83, 91, 96, 97, 116, 124]for identifying source terms of elliptic equations and [25, 123] for computational methods.However, despite a vast literature for inverse problems for elliptic equations, those withobservations on the boundary are not many The practical situation is that the data of aconcerned physical process could be measured only on the boundary (or a part of theboundary) and we have to determine what the process is in the domains or at the other parts

of the boundary For example, in Medicine, to determine the electroencephalogram orelectrocardiogram, people use the voltage and current measured on the skull or chest andthen solve a Cauchy problem for the corresponding elliptic equations, [19, 27, 28, 58]

with c > 0 and a = a(x) ≥ 0 a.e in Ω

The Cauchy problem for second-order linear elliptic equations occurs in many real-worldproblems For example, to figure out what is happening with a field (gravity, electricity,magnetism) in unseen areas, we often have to determine the potential of the field in anexternal part of the mass (charge, current) creating the field, from observations in a part ofthis domain This lead to a problem of extending an analytic vector field in R3or a curve in

R2 to a harmonic vector field outside with determination of its singularities [44] Someliterature reviews of the Cauchy problems for elliptic equations can be found in [28, 44, 54,

61, 94] Generally, the problems are proposed as follows: Let Ω be a domain in Rn, n ≥ 2,with smooth enough boundary Consider the problem of determining u = u(x) such that

a ij u x icos(ν, x i) −X n i=1 4 b i u(x) cos(ν, x i),

(ν, x i) is the angle formed by the outer normal vector of ∂Ω and x i It seems that Lavrent’evwas the first who studied the Cauchy problem for elliptic equations from the viewpoint of ill-posed problems [75, 77–79] and up to now there have been many papers devoted to thisproblems Lavrent’ev established stability estimates of H¨older type for Cauchy problems forthe Laplace equations and then generalized it for second-order elliptic equations Similarresults were given by Landis in [71–73] by using different methods In 1958, Pucci studiedthe conditional stability of the problems with assuming that the solutions are non-negative.After that, Miller [87, 88], Payne [94], Falk and Monk [24], Han [37, 38], Fursikov [30], etc.,used logarithm-convex method; Payne [92,93] presented the weighted function method toachieve stability estimates results for these problems The most popular and effectivemethod for getting stability estimates for ill-posed problems is the Carleman estimatemethod, see [14, 18,20,48,54,108] From these studies, the stability estimate has beenpartly proved for some classes of the Cauchy problems However, in general, the solutions

of the Cauchy problems for elliptic equations are unstable That means, a small perturbation

in the Cauchy data can lead to an arbitrary error in the solution, so, it is very difficult to applynumerical methods to solve them The reason is that besides perturbations in measuring,there are unadvoidable errors due to the discretization process and the round-off errors ofthe computers Therefore, finding stable and effective algorithms for the problems desired.There have been many papers on regularization and approximation methods for Cauchyproblems for elliptic equations so far In 1955, C Pucci [100] suggested a method toapproximate the solution of the Cauchy problem for the Laplace equation in a stable way.Later, in 1956, Lavrent’ev [79] proposed two methods to regularize the Cauchy problems forthe Laplace equation The first based on the Fourier method and the regularization led todetermining the coefficients of the Fourier series However, these methods of Pucci andLavrent’ev were not feasible in practice The second was Lavrent’ev regularization thatmentioned above In 1963, Tikhonov [109] created the so-called Tikhonov regularizationmethod to stabilize ill-posed problems and since then the theory of ill-posed problems hasbeen intensively developed Besides regularization methods mentioned above, many othermethods are proposed including: Quasi-reversibility method [11, 12, 66, 74], Iterativemethod [23, 43, 57]; Backus-Gilbert method [46, 80], Finite difference method [9,103,113],Mollification method [39], Non-local boundary value problem method [85, 115, 119, 121],etc However, there is no universal method for all the Cauchy problem for elliptic equations.Therefore, developing a new method for special classes of elliptic equations is desired

5

In [40], to solve the Cauchy problem

where A : D(A) ⊂ H −→ H is a positive-definite, self-adjoint operator with compact inverse

on a Hilbert space H, the noise level ϵ and T are given positive numbers, H`ao et al

regularized it by the non-local boundary problem



  u tt = Au, 0 < t < γT, uT, u(0) +

αu(γT, uT) = φ, u t(0) = 0,

(0.7)

with γT, u > 1 being given and α > 0 the regularization parameter Furthermore, they suggested

a priori and a posteriori parameter choice rules to yield order-optimal convergence rates.These results are devoted for the abstract equations, hence, to apply them to the Cauchyproblem (0.3) we have to define the solution to it as well the solution to its regularized

al is that they suggested the priori and posteriori strategies of choosing regularizationparameter depending on the perturbation such that the numerical solutions tend to the exactsolution with optimal convergence rate However, the exact solutions defined in [40] areclassic solutions, which demand high smoothness of the data Furthermore, it seems thatnumerical methods for the non-local boundary value problem (0.8) are not developed Wetherefore introduce concept of very weak solution to the Cauchy problem for ellipticequations with Cauchy data in L2(Ω) and very weak solution to the non-local boundary valueproblem (0.8) and then suggest a stable finite difference scheme for it We note that thenotion of very weak solution has been introduced by Ladyzhenskaya for mixed boundaryvalue problems for parabolic and hyperbolic equations [68, 69] and by Lions [82, 83] foranother equations

6The next problem is studied in this thesis is the problem of determining a source term forlinear elliptic equations:

variables, with some more conditions, the uniqueness of solution can be established, see

7e.g [26, 97, 114, 124] In [97] Prilepko proved that the source term of the Poisson equationcan be uniquely determined if it is independent of one of the variables and is monotone

Vabishchevich [114] also proved that determining the source term is unique but it mustsatisfy some curious conditions In [124], W Yu considered the problem of determining (u, f)from

1 h(x, y) is strictly positive in Q

2 h is monotonic with respect to y, that is ∂ y h(x, y) ≥ 0 and ∂ y h ̸= 0

W Yu [124] proved that results under the weaker assumptions Specifically, Yu proved that

if the problem satisfied one of following conditions:

1 h(x, y) > 0 almost everywhere in Q and ∂ y h(x, y) ≥ 0 and ∂ y h(x, y) ̸= 0, 2

h(x, y) ≥ α > 0 and has a derivative bounded from below with respect to y, 3 f

does not depend upon y, that is ∂ y f = 0 when f > 0,

then the existence in the above-mentioned inverse problem in the same space as that in theinvestigation of uniqueness is obtained, using the theory of solvable operators betweenBanach spaces Furthermore, he obtained also the continuous dependence of the solution

to the inverse problem on data Hence, problem (0.14)-(0.15) is well-posed in sense ofHadamard He also noted that the results in [124] could be applied for the similar problemswith Dirichlet or Neumann boundaries

8

In general, however, the source determination problem for elliptic equations is an

ill-posed problem So, we need some appropriate definition to ensure the uniqueness ofsolution in certain sense and regularization methods to approximate it by a well-posedproblem Since in verse problem (0.9)-(0.10) may have many solutions, we introduce the so-called f ∗-minimum norm solution which is nearest the a-priori f ∗among all the least squaressolutions to it We will give a specific example to show that the solution to the above inversesource problem is unstable Then the problem is reformulated in an abstract setting andusing Tikhonov reg ularization for solving it This leads to an optimal control problem withcontrol constraints To solve the problem numerically we discretize the regularized problem

by finite-dimensional problems based on Hinze’s variational discretization concept in optimalcontrol [45] This is a new discretization concept for optimal control problems with controlconstraints Its key feature is not to discretize the space of admissible controls but toimplicitly utilize the first order optimality conditions and the discretization of the state andadjoint equations for the discretization of the control For discrete controls obtained in thisway an optimal error estimate is proved However, we go a little further than that for optimalcontrol by Hinze, namely, we suggest a choice of the regularization parameter depending onthe noise level in the observation data and the discretization mesh size which yields theconvergence of the solution to the discretized regularized problem to the solution of thecontinuous inverse problem as these quantities tend to zero This is one of the maincontributions of Chapter 2 Furthermore, with this choice a convergence rate is alsoestablished

As problem (0.14)-(0.15) has a special form, its solution can be represented by Fourierseries Therefore, instead of using Tikhonov regularization method for it, in Chapter 3 weshall apply the standard truncated Fourier series method There we prove error estimates ofthe method and present some numerical examples for showing its efficiency

This thesis is organized as follows:

Chapter 1 is devoted to the Cauchy problem for elliptic equations There we introduce anew concept of very weak solution to the Cauchy problem for elliptic equations and for anon-local boundary value problem which regularizes the Cauchy problem We discretize thenon-local boundary problem by the finite difference method and prove the stability of thescheme and its convergence We present some numerical examples for showing theefficiency of the regularization method

In Chapter 2, we study the source identification problem for elliptic equations with observation on the boundary We formulate the inverse problem as an operator equation andregularize it by the Tikhonov regularization method To discretize the problem, we followHinze’s idea on variational discretization and suggest a rule of choosing the regularization

9parameter depending on the noise level in the observation data and the discretization meshsize which yields the optimal convergence rate This abstract result is applied to the finiteelement method for numerical solving the source identification problem Some numerical

examples are presented for showing the efficiency of the method

Chapter 3 is devoted to the problem of determining a term in the right-hand side ofelliptic equation with constant and variable coefficients in a cylinder from boundaryobservations Based on the special form of the considered equation in a cylinder, thesolution of the direct and inverse problems can be represented by the Fourier series As theproblem is ill-posed, we regularize it by truncating the Fourier series We prove errorestimates of the method and present some numerical examples for showing its efficiency

10

Chapter 1

The Cauchy problem for

elliptic equations

In this chapter we will regularize the Cauchy problem (1.3) by a non-local boundaryproblem for elliptic equations After introducing the definition of the very weak solution tothese problems, we prove the well-posedness of the non-local boundary problem anddiscretize it by the finite difference method as well as prove the stability of the scheme andthus its convergence Some numerical examples are given for illustrating the efficiency ofthe method This chapter is written on the basis of the paper [41].(1)

1.1 Problem settings and the non-local boundary

problem regularization method

Let Ω be a bounded domain in Rnand T be a given positive number Let a ij ∈ C(Ω), a

∈ L ∞(Ω), i, j = 1, , n such that

(1.1)

the very weak solution to a Cauchy problem for an elliptic equation Journal of Inverse and Ill-Posed Problems 26(6), 835–857

11with c > 0 and a = a(x) ≥ 0 a.e in Ω In this chapter we consider following Cauchy problem



t < T, u| ∂Ω = 0, 0 < t < T,  u(x, 0) = φ, φ ∈ L2(Ω), u t(x, 0) = 0, x ∈ Ω

(1.2)

Let ϵ be the noise level of the Cauchy condition, problem (1.2) with noise is: Finding u(x, t) such that u tt +Xn i,j=1 (a ij u x i)x j − au = 0, x ∈ Ω, 0 <

v α(t) → u(t) with the optimal convergence rate as following results:

Denote by v α(x, t) ∈ C2((0, T), L2(Ω)) ∩ C([0, T]) the solution to (1.5) with the regu larization parameter α, u(., t) the solution to (1.2) and ∥ · ∥ := ∥ · ∥ L2(Ω)

Theorem 1.1.1 (A priori parameter method choice) Suppose that γT, u > 1 If there

exists a positive number E such that

∥u(·, T)∥ ≤ E, (1.6)

then with α = ( ϵ E)γT, u we have

∥u(·, t) − v α(·, t)∥ ≤ Cϵ1− tT E tT , ∀t ∈ [0, T] (1.7) Here, C is a finite constant depending only on γT, u and T

Theorem 1.1.2 (A posteriori parameter methodξ choice) Let ϵ < ∥φ∥ andξ γ >

1 Taking τ ∗ > 1 such that 0 < τ ∗ ϵ < ∥φ∥, if there exists a positive number E

Remark 1.1.1 Suppose that A admits an orthonormal system of eigenfunctions v k ∈

L2(Ω): Av k = λ k v k such that 0 < λ1 ≤ λ2 ≤ , lim

C ℓ(Ω), a ∈ C ℓ−1(Ω) with a positive integer ℓ, then for u(x, T) = Φ such that

13

• Φ ∈ H ℓ+1(Ω),

• Φ, AΦ, , A[ℓ

2 ]Φ ∈

yields the constraint (1.10) for β = ℓ − 1

Remark 1.1.3 (i) The results of Theorem 1.1.1 are valid for γT, u = 1 (ii) The two above

parameter choosing methods are of optimal order (iii) The a posterior parameter

choice in Theorem 1.1.2 is equivalent to the equation α∥v α(·, γT, uT)∥ = τ ∗ ϵ

The results of [40] are devoted to abstract equations, hence to apply them to the problem (1.3), we have to define the solution to it as well the solution to (1.5) Furthermore, itseems that numerical methods for the non-local boundary value problem (1.5) have notdeveloped We therefore introduce a concept of very weak solution to the Cauchy problemfor elliptic equations with Cauchy data in L2and very weak solution to the non-local boundary value problem (1.5) and then suggest a stable finite difference scheme for it We notethat Ladyzhenskaya in [68, 69] and Lions in [82, 83] has been introduced the notion of veryweak equations for mixed boundary value problems for parabolic and hyperbolic for otherequations

1.2 The very weak solution to the Cauchy problem for

elliptic equations

Let D be a bounded domain in Rk with boundary ∂D consisting of two non-intersectedparts S0 and S1 Let the functions b ij ∈ C(D), i, j = 1 , k, b ∈ L ∞(D) satisfying thecondition of uniform ellipticity

It is common that the solution to the first equation in the Cauchy problem (1.11) issought in the space H1(D) Therefore, if S0 is not too rough, φ must be in H 1/2(S0).However, in practice this Cauchy datum should be in L2(S0) and so it is reasonable to find u

in L2(D) But, in this case there is no trace of u on S0 To give a precise definition for thesolution in L2(D), multiplying the both side of the first equation of (1.11) by a smooth function

ψ defined on D then formally taking integration by parts, we get

function u ∈ L2(D) such that (1.16) is satisfied for all p ∈ P With this definition, all items inthe both sides of (1.16) have a meaning, since from the assumption ψ ∈ H1(D) we have ψ|

S0 ∈ H 1/2(S0) and since b ij ∈ C(D), ∂ψ/∂N ∈ H −1/2(S0) Furthermore, if there exists a veryweak solution to (1.11), it is unique In fact, if for the Cauchy data φ there are

two very weak solutions u1 and u2 to (1.11), then

Z

0 =

for all p ∈ P Since P is dense in L2(D), it follows that u1 − u2 = 0 in L2(D)

Now we turn to the Cauchy problem:

P, a dense set in L2(Ω × (0, T)), there exists a unique solution in H1(Ω × (0, T)) to problem(1.19) Then, similarly to problem (1.11), we can conclude that all items in the both sides of(1.20) have a meaning and the very weak solution to (1.18) is unique We now present arepresentation to the solution In doing so, consider the eigenvalue problem





−Pni,j=1(a ij u x i)x j + au = λu, x ∈ Ω, u| ∂Ω = 0.(1.21)

The solution to this problem is a function u ∈ H10(Ω) such that

with φ k =RΩφ(x)v k(x)dx and φ 1k =RΩφ1(x)v k(x)dx converges in L2(Ω) for almost t ∈ (0, T),

it is the very weak solution to (1.18) It is clear that in this case φ1 and φ2 are very smooth(see below), however, as we see from the condition (0.2), in general the Cauchy data φ and

φ1 are not independently chosen Hence, a sufficient and necessary condition for thesolvability of the Cauchy problem (1.18) is not clear to us

17When f = 0 and φ1 = 0, we have, u(x, t) = X∞ k=1

1.3 The well-posedness of the non-local boundary value

problem for elliptic equations

In this section we analyze the solvability of problem (1.5) Since the parameter γT, u plays

no role in this part, we suppose that it equals to 1 As for the Cauchy problem (1.18), if thesolution to this problem is sought in H1(Ω×(0, T)), then u(·, 0) = φ(·) must be in H 1/2(Ω),while this datum is given in L2(Ω) Therefore, we have to introduce a new definition for asolution to this problem

1.3.1 The very weak solution

From the formal formula (1.14) we introduce the following notion to a solution to problem(1.5)

Definition 1.3.1 A function v ∈ C([0, T], L2(Ω)) is called the very weak solution to problem(1.5) if it satisfies

for all ψ ∈ H1(Ω × (0, T)) such that ψ tt − Aψ ∈ L2(Ω × (0, T)) and ψ| ∂Ω = ψ(x, T) = 0

Theorem 1.3.1 Let φ be a function in L2(Ω) Then, the exists a unique very weak solution to problem (1.5) which can be represented in the form

Consider the following Dirichlet problems

Z T

0

Z

vp˜ (x, t)dxdt = 0, (1.34) Ω

for all p ∈ L2(Ω × (0, T)) It follows that v˜ = 0 The uniqueness has been proved

Moreover, the solution v(x, t) defined in (1.27) is stable Indeed,

reasonable to study when there exists a classical solution to it

1.3.2 The classical solution

A classical solution to problem (1.5) is a function u twice continuously differentiable in Ω

× (0, T) and satisfies the problem pointwise For guaranteeing the existence of a classicalsolution, some regularity conditions on the coefficients and data of problem (1.5) arerequired Following [50] we suppose that Ω ⊂ Ω, an open bounded domain in e

Rnand

a ij ∈ C (1,µ)(Ω) e , a ∈ C (0,µ)(Ω) e , µ > 0, (1.35) which means that a ij are differentiable in Ω and their first derivatives satisfy the H¨older e

condition with exponent µ in any closed subset of Ω, the function e

a ∈ C(Ω) and satisfiesthe e H¨older condition with exponent µ in any closed subset of Ω In [51,52], Il’in and

Shishmarev e proved that if the Dirichlet problem for the Laplace equation is solvable in Ωfor arbitrary continuous boundary data on ∂Ω and the condition (1.35) is satisfied, then theeigenfunctions in H10(Ω) of problem (1.21) are classical eigenfunctions that means that theeigenfunctions v k are twice continuously differentiable and satisfy (1.21) in the classicalsense Furthermore, we have the following results concerning the Fourier series generated

by eigenfunctions v k:

Lemma 1.3.2 [50, Lemma 1, p 123–124] Let Ω ⊂ Ωe

, an open bounded domain in

Rn , and the conditions (1.35) be satisfied Then the series from the eigenfunctions of problem (1.21) of the form

Lemma 1.3.3 [50, Lemma 5, p 139] Let Ω be a connected, bounded domain in R n and a ij ∈ C ℓ(Ω), a ∈ C ℓ−1(Ω) with a positive integer ℓ Furthermore, suppose that the function Φ satisfies

• Φ ∈ H ℓ+1(Ω),

• Φ, AΦ, , A[ℓ

2 ]Φ ∈ H10(Ω) Then

Theorem 1.3.4 Assume that the Dirichlet problem for the Laplace equation is solv

able in Ω for arbitrary continuous boundξ ary dξ ata on ∂Ω and the condition (1.35)

is satisfiedξ Let φ ∈ H[n]+1(Ω) and

φ, Aφ, , A[n]+1φ ∈ H10(Ω), (1.39)

the coefficients a ij ∈ C[n]+2(Ω) andξ the coefficient a ∈ C[n]+1(Ω) Then the series

(1.27) is uniformly convergent in Ω × (0, T), its second order derivatives v tt and v x i x j are uniformly convergent in any closed subdomain of Ω × (0, T) The series (1.27) is the classical solution to the problem (1.5)

Proof 1) The uniform convergence of the series (1.27) in Ω × (0, T) Since

|φ k |2λ k 2

Since the conditions of Lemmas 1.3.2, 1.3.3 are satisfied with ℓ = [n2] the two factors in thelast product converge uniformly in Ω × (0, T) Hence this part is proved It follows furtherthat the series (1.27) satisfies the condition v(x, 0) + αv(x, T) = φ(x) pointwise

2) The uniform convergence of the second derivatives of v(x, t) The formal secondderivatives with respect to t and to x i and x j of the series (1.27) have the form

In case Ω = (0, L) and Au = u xx, the eigenfunctions and eigenvalues of the one 22dimensional problem (1.21) correspondingly

are r2

In case Ω = (0, L1) × (0, L2) and Au = ∆u := u x1x1 + u x2x2, the eigenfunctions and

eigenvalues of the two-dimensional problem (1.21) correspondingly are r4

1.4 Finite difference scheme for the non-local

boundary value problem

In this section, we introduce a finite difference scheme for problem (1.5) We restrictourselves for two and three-dimensional problems and for Ω being either the interval (0, L)

or the parallelepiped (0, L1) × (0, L2)

23Following [102], we subdivide the domain [0, L] × [0, γT, uT] into cells by the rectangular

uniform grid specified by

ω h = {x i = ih, h > 0, 1 ≤ i ≤ M − 1, Mh = L},

ω τ = {t j = jτ, τ > 0, 1 ≤ j ≤ N − 1, Nτ τ = γT, uT},

with M and Nτ being the numbers of grid points in [0, l] and [0, γT, uT], respectively For a

function y defined on this grid we introduce the standard approximations to its derivatives:

The interior of ω h is denoted by ω h and its boundary by Ξh We denote by L2(ω h) the space

of grid functions given in ω h with the scalar product

1.4.1 The three-dimensional problem

Let Ω = (0, L1) × (0, L2) Consider the three-dimensional non-local boundary value

(1.52)

Here, x = (x1, x2) and Au := −(a1u x1)x1 − (a2u x2)x2 + au, γT, u > 1 The coefficients a1, a2 and c

belong to C(Ω) and satisfy the condition of uniform ellipticity (1.1):

0 < c1 ≤ a1(x), a2(x) ≤ c2, 0 ≤ c3 ≤ a(x) ≤ c4 for all x ∈ Ω (1.53) for some given positive constants c1, c2, c3 and c4

To discretize φ ∈ L2(Ω) we either take its average over a cell or we first mollify it by asmooth function and then take its values at nodes of ω¯h Since problem (1.52) is well-posed, the mollification process does not magnify the error in φ Hence we take conventionthat φ ∈ C(Ω) and discretize problem (1.52) by

We rewrite problem (1.55) into the operator equation A h v = λv by defining the operator

A h v = −(a1v˜x¯1)x1 − (a2v˜x¯2)x2 + av˜, where v ∈ L2(ω h), v˜ ∈ L2h0and v(x) = v˜(x) for x ∈

ω h

Define the discrete Laplace operator L h v = −v˜x¯1x1 − v˜x¯2x2, where v ∈ L2(ω h), v˜ ∈

L2h0 and v(x) = v˜(x) for x ∈ ω h We have the following result:

Lemma 1.4.2 [104, Lemma 19, p 19] The operator A h is self-adjoint in L2(ω h) and the following estimates hold:

(c1 + c3/∆)(L h v, v)L2(ω h) ≤ (A h v, v)L2(ω h) ≤ (c3 + c4/δ)(L h v, v)L2(ω h), (1.56) (c1δ + c3)(v, v)L2(ω h) ≤ (A h v, v)L2(ω h) ≤ (c2∆ + c4)(v, v)L2(ω h), (1.57) where

δ =X2 2M i ≥8L21+8L22, ∆ = X2 h2

isin2

π

4

i=1

It follows that the eigenvalues λ kℓ, k = 1, , M1 − 1; ℓ = 1, , M2 − 1, of prob lem(1.55) are positive and the corresponding eigenfunctions v kℓ which are supposed to beorthonormalized in L2(ω h)

Now, we look for a solution y(t, x) to the finite difference system (1.54) in the form

det A = α det A Nτ + (−1)Nτ +2(−1)Nτ = α det A Nτ + 1

We prove that A Nτ is symmetric and positive definite We have A Nτ = A 1Nτ + A 2Nτ , with

Applying the Laplace expansion for the first row of A Nτ , we get

det A 1Nτ = det H N−1 − det H N−2 , (1.64) with

Applying the Gershgorin theorem(2), we have for the eigenvalues of A 1Nτ the estimate |λ1

− 1| ≤ 1, |λ i − 2| ≤ 2, i = 2, 3, , Nτ , |λ Nτ +1 − α| ≤ α and hence 0 ≤ λ i with i =

1, , Nτ + 1 If λ i = 0 were an eigenvalue of A 1Nτ , it would follow that det A 1Nτ = 0 It is acontradiction to the equality det A 1Nτ = 1 Thus, A 1Nτ is positive definite Clearly, A 2Nτ is semi-positive definite Hence A Nτ is a positive definite matrix Since det A = α det A Nτ +1, we havedet A ̸= 0 So, equation (1.63) is uniquely solvable

Next, we prove that the solution to the discrete problem (1.54) is stable in L2(ω h × ω τ ).For this purpose, we apply the elimination method [102, p 9–14] to the system (1.62) bysetting

µ i = α i+1 µ i+1 + β i+1

k=1,k̸=i

|a ik |