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The paper gives an explicit representation of the Green’s function of the Dirichlet boundary value problem for the polyharmonic equation in the unit ball. The solution of the homogeneous Dirichlet problem is found. An example of solving the homogeneous Dirichlet problem with the simplest polynomial right-hand side of the polyharmonic equation is given.

The interesting connection between random walks and electric networks is the subject of the book 'Random walks and electric networks' written by Peter G. Doyle and J. Laurie Snell. This book was published in 1984 by the Mathematical Association of America and it is now freely redistributable under the terms of the GNU General Public License. Since thebook is freely available on the internet, a solutions manual willbe a useful companion for the interested reader. Thismanual hasthe solutions of the exercises which appear in Part Iof the book, i.e.Random walks on finite networks. The solutions manual also contains three appendices which the reader can consult and a short bibliography. &nbsp

Dunkl operators are a family of commuting differential–difference operators associated with a finite reflection group. These operators play a key role in the area of harmonic analysis and theory of spherical functions. We study the solution of the inhomogeneous Dunkl polyharmonic equation based on the solutions of Dunkl–Possion equations. Furthermore, we construct the solutions of Dirichlet and Neumann boundary value problems for Dunkl polyharmonic equations without invoking the Green’s function.

Formal solution of the Dirichlet problem in a ball for a non-homogeneous ultrahyperbolic equation with a polynomial right-hand part. In the paper, a formal solution of the Dirichlet problem in a ball for a nonhomogeneous ultrahyperbolic equation with a polynomial right-hand side is found. The solution construction procedure is based on the apparatus of spherical functions and the theory of Gauss's hypergeometric equation. At the same time, the sought function and the known right-hand side of the investigated equation are expanded into a Fourier series by spherical harmonics, which are eigenfunctions of the Laplace-Beltrami operator. The specified decomposition allows to reduce the original ultrahyperbolic equation to an ordinary inhomogeneous differential equation of the second order. The corresponding homogeneous equation is transformed by substitution into a hypergeometric Gaussian equation, the study of which consists in a detailed analysis of the so-called degenerate case, when the solution can be expressed in terms of any two of Kummer's 24 series. Difficulties in proving solution smoothness are due to the fact that each subsequent term of the formal series is expressed in terms of the previous one using cumbersome recurrence relations., В роботі знайдено формальний розв’язок задачі Діріхле у кулі для неоднорідного ультрагіперболічного рівняння з поліноміальною правою частиною. Процедура побудови розв’язку базується на апараті сферичних функцій та теорії гіпергеометричного рівняння Гаусса. При цьому шукана функція та відома права частина досліджуваного рівняння розкладаються в ряд Фур’є за сферичними гармоніками, які є власними функціями оператора Лапласа-Бельтрамі. Зазначене розкладання дозволяє привести вихідне ультрагіперболічне рівняння до звичайного неоднорідного диференціального рівняння другого порядку. Відповідне однорідне рівняння за допомогою підстановки перетворюється на гіпергеометричне рівняння Гаусса, дослідження якого полягає у детальному аналізі так званого виродженого випадку, коли розв’язок може бути виражений через будь-які два з 24 рядів Куммера. Складнощі доведення гладкості розв’язку задачі Діріхле для ультрагіперболічного рівняння пов’язані з тим, що кожен наступний член формального ряду виражається через попередній за допомогою громіздких рекурентних співвідношень.

In this paper we develop a potential theory for strongly degenerate parabolic operators of the form L : = ∇ X ⋅ ( A ( X , Y , t ) ∇ X ) + X ⋅ ∇ Y − ∂ t , in unbounded domains of the form Ω = { ( X , Y , t ) = ( x , x m , y , y m , t ) ∈ R m − 1 × R × R m − 1 × R × R | x m > ψ ( x , y , y m , t ) } , where ψ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator L . Concerning A = A ( X , Y , t ) we assume that A is bounded, measurable, symmetric and uniformly elliptic (as a matrix in R m ). Beyond the solvability of the Dirichlet problem and other fundamental properties our results include scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures. All of our estimates are translation- and scale-invariant with constants only depending on the constants defining the boundedness and ellipticity of A and the Lipschitz constant of ψ. Our results represent a version, for operators of Kolmogorov type with bounded, measurable coefficients, of the by now classical results of Fabes and Safonov, and several others, concerning boundary estimates for uniformly parabolic equations in (time-dependent) Lipschitz type domains.

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,\mu_X)$ satisfying a $2$-Poincar\'e inequality. Given a bounded domain $\Omega\subset X$ with $\mu_X(X\setminus\Omega)>0$, and a function $f$ in the Besov class $B^\theta_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^\theta_{2,2}(X)$ such that $u=f$ in $X\setminus\Omega$ and $\mathcal{E}_\theta(u,u)\le \mathcal{E}_\theta(h,h)$ whenever $h\in B^\theta_{2,2}(X)$ with $h=f$ in $X\setminus\Omega$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally H\"older continuous on $\Omega$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups., Comment: 42 pages, comments welcome, submitted. Revision to add crucial references and attributions to the introduction

We extend the theory of lower and upper solutions to planar systems of ordinary differential equations with separated boundary conditions, both in the well-ordered and in the non-well-ordered cases. We are able to deal with general Sturm–Liouville boundary conditions in the well-ordered case, and we analyze the Dirichlet problem in the non-well-ordered case. Our results apply in particular to scalar second order differential equations, including those driven by the mean curvature operator. Higher dimensional systems are also treated, with the same approach.

This thesis aims to provide an introduction to the field of potential theory at an undergraduate level, by studying an important mathematical problem in the field, namely the Dirichlet problem. By examining the historical development of different methods for solving the problem in increasingly general contexts, and the mathematical concepts which were established to support these methods, the aim is to provide an overview of various basic techniques in the field of potential theory, as well as a summary of the fundamental results concerning the Dirichlet problem in Euclidean space. Målet med detta arbete är att vara en introduktion på kandidatnivå till ämnesfältet potentialteori, genom att studera ett viktigt matematiskt problem inom potentialteori, Dirichletproblemet. Genom att undersöka den historiska utvecklingen av olika lösningsmetoder för problemet i mer och mer allmänna sammanhang, i kombination med de matematiska koncepten som utvecklades för att användas i dessa lösningsmetoder, ges en översikt av olika grundläggande tekniker inom potentialteori, samt en sammanfattaning av de olika matematiska resultaten som har att göra med Dirichletproblemet i det Euklidiska rummet.

We consider quite general fully nonlinear mixed Cauchy-Dirichlet problems with a Caputo derivative D-alpha with respect to the time variable and a in (0, 2). Under natural conditions, we show the existence of a local solution u such that D(alpha)u and the second order space derivatives D-xi,D-x j u belong to the class C-alpha theta/2,C-theta ([0, T] x (Omega) over bar), for some T positive, with theta is an element of(0, 1). Moreover, we show the uniqueness of global solutions in the same class of functions.

Let $\Delta_k$ be the Dunkl Laplacian relative to a fixed root system $\mathcal{R}$ in $\mathbb{R}^d$, $d\geq2$, and to a nonnegative multiplicity function $k$ on $\mathcal{R}$. Our first purpose in this paper is to solve the $\Delta_k$-Dirichlet problem for annular regions. Secondly, we introduce and study the $\Delta_k$-Green function of the annulus and we prove that it can be expressed by means of $\Delta_k$-spherical harmonics. As applications, we obtain a Poisson-Jensen formula for $\Delta_k$-subharmonic functions and we study positive continuous solutions for a $\Delta_k$-semilinear problem.

In this paper, we investigate the Monge–Ampere equation $$\text{ det }D^2u=f $$ in $${\mathbb {R}}^n_+$$ , where f is bounded, positive and $$f(x)=1+O(|x|^{-\beta })$$ for some $$\beta >2$$ at infinity. If u is a quadratic polynomial on $$\{x_n=0\}$$ and satisfies $$ \mu |x|^2\le u\le \mu ^{-1}|x|^2$$ for some $$00$$ is some constant depending only on $$\beta $$ and n. Meanwhile, the existence and uniqueness of viscosity solutions of the Dirichlet problem with prescribed asymptotic behavior at infinity will be concerned. The condition $$\beta >2$$ is sharp.

We provide parameter dependent version of the Browder–Minty Theorem in case when the solution is unique utilizing different types of monotonicity and compactness assumptions related to condition (S)2. Potential equations and the convergence of their Euler action functionals is also investigated. Applications towards the dependence on parameters for both potential and non-potenial nonlinear Dirichlet boundary problems are given.

We consider a prototypical nonlinear parabolic equation whose flux has three distinguished features: it is nonlinear with respect to both the unknown and its gradient, it is homogeneous, and it depends only on the direction of the gradient. For such equation, we obtain existence and uniqueness of entropy solutions to the Dirichlet problem, the homogeneous Neumann problem, and the Cauchy problem. Qualitative properties of solutions, such as finite speed of propagation and the occurrence of waiting-time phenomena, with sharp bounds, are shown. We also discuss the formation of jump discontinuities both at the boundary of the solutions’ support and in the bulk.

In the half-plane {−∞ < x < +∞} × {0 < y < +∞}, the Dirichlet problem is considered for differential-difference equations of the kind $$ {u}_{xx}+\sum_{k=1}^m{a}_k{u}_{xx}\left(x+{h}_{k,}y\right)+{u}_{yy}=0 $$ , where the amount m of nonlocal terms of the equation is arbitrary and no commensurability conditions are imposed on their coefficients a1, . . . , am and the parameters h1, . . . , hm determining the translations of the independent variable x. The only condition imposed on the coefficients and parameters of the studied equation is the nonpositivity of the real part of the symbol of the operator acting with respect to the variable x. Earlier, it was proved that the specified condition (i.e., the strong ellipticity condition for the corresponding differential-difference operator) guarantees the solvability of the considered problem in the sense of generalized functions (according to the Gel’fand–Shilov definition), a Poisson integral representation of a solution was constructed, and it was proved that the constructed solution is smooth outside the boundary line. In the present paper, the behavior of the specified solution as y → +∞ is investigated. We prove the asymptotic closeness between the investigated solution and the classical Dirichlet problem for the differential elliptic equation (with the same boundary function as in the original nonlocal problem) determined as follows: all parameters h1, . . . , hm of the original differential-difference elliptic equation are assigned to be equal to zero. As a corollary, we prove that the investigated solutions obey the classical Repnikov–Eidel’man stabilization condition: the solution stabilizes as y → +∞ if and only if the mean value of the boundary function over the interval (−R,+R) has a limit as R → +∞.

We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball. In this paper, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant. We derive an integral identity for the fourth order Dirichlet problem and a nonlinear weighted trace inequality, and the combination of them results in a quantitative stability estimate which measures the deviation of a domain from a ball in terms of the perturbation of the boundary value.

In the junction Ω \Omega of several semi-infinite cylindrical waveguides, the Dirichlet Laplacian is treated whose continuous spectrum is the ray [ λ † , + ∞ ) [\lambda _\dagger , +\infty ) with a positive cutoff value λ † \lambda _\dagger . Two different criteria are presented for the threshold resonance generated by nontrivial bounded solutions to the Dirichlet problem for the Helmholtz equation − Δ u = λ † u -\Delta u=\lambda _\dagger u in Ω \Omega . The first criterion is quite simple and is convenient to disprove the existence of bounded solutions. The second criterion is rather involved but can help to detect concrete shapes supporting the resonance. Moreover, the latter distinguishes in a natural way between stabilizing, i.e., bounded but nondecaying solutions, and trapped modes with exponential decay at infinity.

We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy-Sobolev parabolic equation, in the energy space. The aim of this paper is to determine a necessary and sufficient condition on initial data below or at the ground state, under which the behavior of solutions is completely dichotomized. More precisely, the solution exists globally in time and its energy decays to zero in time, or it blows up in finite or infinite time. The result on the dichotomy for the corresponding Dirichlet problem is also shown as a by-product via comparison principle., 49 pages

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Holder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, $$ where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that $$ {\partial}_{\overline{z}}f=g $$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincare problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $$ {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). $$ On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

Context. The question of constructing a method of two-sided approximations for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation based on the use of the Green’s functions method is considered. The object of research is the first boundary value problem (the Dirichlet problem) for a second-order semilinear elliptic equation. Objective. The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order semilinear elliptic equations based on the use of the Green’s functions method and to study its work in solving test problems. Method. Using the Green’s functions method, the initial first boundary value problem for a semilinear elliptic equation is replaced by the equivalent Hammerstein integral equation. The integral equation is represented in the form of a nonlinear operator equation with a heterotone operator and is considered in the space of continuous functions, which is semi-ordered using the cone of nonnegative functions. As a solution (generalized) of the boundary value problem, it was taken the solution of the equivalent integral equation. For a heterotone operator, a strongly invariant cone segment is found, the ends of which are the initial approximations for two iteration sequences. The first of these iterative sequences is monotonically increasing and approximates the desired solution to the boundary value problem from below, and the second is monotonically decreasing and approximates it from above. Conditions for the existence of a unique positive solution of the considered Dirichlet problem and two-sided convergence of successive approximations to it are given. General guidelines for constructing a strongly invariant cone segment are also given. The method developed has a simple computational implementation and a posteriori error estimate that is convenient for use in practice. Results. The method developed was programmed and studied when solving test problems. The results of the computational experiment are illustrated with graphical and tabular informations. Conclusions. The experiments carried out have confirmed the efficiency and effectiveness of the developed method and make it possible to recommend it for practical use in solving problems of mathematical modeling of nonlinear processes. Prospects for further research may consist the development of two-sided methods for solving problems for systems of partial differential equations, partial differential equations of higher orders and nonstationary multidimensional problems, using semi-discrete methods (for example, the Rothe’s method of lines).

For a Jordan domain with sufficiently smooth boundaries, the solution to the Dirichlet problem for second order skew-symmetric strongly elliptic system with constant coefficients and regular enough boundary data is constructed in the form of a power series of a small parameter describing the perturbation of the given system from the Laplace one. The coefficients of this series are the functions that are determined sequentially as solutions to special Dirichlet problems for the usual Laplace and Poisson equations. The obtained series converges uniformly in the closure of the domain under consideration.

The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation is analyzed. With the focus on non-homogeneous boundary data, two approaches are offered: one is based on the theory of hyperbolic systems, while the other one uses the theory of operator semigroups. Both methods have in common that known results for the Dirichlet problem for the second-order wave equation are instrumental. The obtained interior and boundary regularity estimates are optimal.

In this paper, we study the Dirichlet problem of elliptic systems $$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{}-\Delta \mathbf {u}=\mathbf {g}(\mathbf {u})&{}\quad &{}\text{ in } \ \Omega ,\\ &{}\mathbf {u}=\mathbf {\varrho } \varvec{\mu }&{}\quad &{}\text{ on } \ \partial \Omega , \end{aligned} \end{array}\right. } \end{aligned}$$ where $$\varrho \ge 0$$ , $$\Omega $$ is an open bounded $$C^2$$ domain in $$\mathbb {R}^N$$ with $$N\ge 2$$ , and $$\mathbf {u}$$ , $$\mathbf {g}(\mathbf {u})$$ , $$\varvec{\mu }$$ are nonnegative vector-valued functions. We obtain the existence of weak positive solutions for the systems. In the special case $$\mathbf {g}(\mathbf {u})=|\mathbf {u}|^{p-1}\mathbf {u}$$ with $$p>1$$ , we shall give a better description about the positive solutions including the priori estimate, regularity, existence and nonexistence.

Исследованы возможности численного метода коллокации и наименьших квадратов (КНК) на примерах кусочно-полиномиального решения задачи Дирихле для уравнений Пуассона и типа диффузии-конвекции с особенностями в виде больших градиентов и разрыва решения на границах раздела двух подобластей. Предложены и реализованы новые hp-варианты метода КНК, основанные на присоединении внутри области малых и/или вытянутых нерегулярных ячеек, отсекаемых криволинейной границей раздела от исходных прямоугольных ячеек сетки, к соседним самостоятельным ячейкам. Выписываются с учетом особенности условия согласования между собой кусков решения в ячейках, примыкающих с разных сторон к границе раздела. Проведено сравнение результатов, полученных методом КНК и другими высокоточными методами. Показаны преимущества и достоинства метода КНК. Для ускорения итерационного процесса применены современные алгоритмы и методы: предобуславливание; свойства локальной системы координат в методе КНК; ускорение, основанное на подпространствах Крылова; операция продолжения на многосеточном комплексе; распараллеливание. Исследовано влияние этих способов на количество итераций и время расчетов при аппроксимации полиномами различных степеней. The capabilities of the numerical least-squares collocation (LSC) method of the piecewise polynomial solution of the Dirichlet problem for the Poisson and diffusion-convection equations are investigated. Examples of problems with singularities such as large gradients and discontinuity of the solution at interfaces between two subdomains are considered. New hp-versions of the LSC method based on the merging of small and/or elongated irregular cells to neighboring independent cells inside the domain are proposed and implemented. They cut off by a curvilinear interface from the original rectangular grid cells. Taking into account the problem singularity the matching conditions between the pieces of the solution in cells adjacent from different sides to the interface are written out. The results obtained by the LSC method are compared with other high-accuracy methods. Advantages of the LSC method are shown. For acceleration of an iterative process modern algorithms and methods are applied: preconditioning, properties of the local coordinate system in the LSC method, Krylov subspaces; prolongation operation on a multigrid complex; parallelization. The influence of these methods on iteration numbers and computation time at approximation by polynomials of various degrees is investigated.

Some theorems concerning the compact classes of homeomorphisms with hydrodynamic normalization, which are solutions of the Beltrami equation and the characteristics of which are compactly supported and satisfy certain constraints of the theoretical-set type, have been proved. As a consequence, we obtained results on the compact classes of solutions of the corresponding Dirichlet problems considered in a certain Jordan domain.

Consider the Dirichlet problem for Laplace’s/Poisson’s equation in a bounded simply-connected domain S . The source function q ln | P Q * ¯ | is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by q ln | P Q * ¯ | as the source node Q * is located near the boundary Γ ( = ∂ S ) . So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node Q * located near Γ . In this paper, an additional FS as, d 0 ln | P Q 0 ¯ | , is added to the original source nodes in the traditional MFS, and the point charge d 0 ( = q ) and the source node Q 0 are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node Q * is located inside S but near Γ , both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function q ln | P Q * ¯ | as the source node Q * is located near Γ .

For the semilinear Dirichlet problem Δ u + g ( u ) = f ( x ) for x ∈ Ω , u = 0 on ∂ Ω decompose f ( x ) = μ 1 φ 1 + e ( x ) , where φ 1 is the principal eigenfunction of the Laplacian with zero boundary conditions, and e ( x ) ⊥ φ 1 in L 2 ( Ω ) , and similarly write u ( x ) = ξ 1 φ i + U ( x ) , with U ⊥ φ 1 in L 2 ( Ω ) . We study properties of the solution curve ( u ( x ) , μ 1 ) ( ξ 1 ) , and in particular its section μ 1 = μ 1 ( ξ 1 ) , which governs the multiplicity of solutions. We consider both general nonlinearities, and some important classes of equations, and obtain detailed description of solution curves under the assumption g ′ ( u ) λ 2 . We obtain particularly detailed results in case of one dimension. This approach is well suited for numerical computations, which we perform to illustrate our results.

In this paper we study the existence of a positive solution for a nonlinear Dirichlet problem driven by the p-Laplacian and a reaction which has the competing effects of a singular term and of a convection (gradient dependent) term. Using the “frozen” variable method and eventually the Leray-Schauder Alternative Theorem, we show that the problem has a positive smooth solution. We mention that on the gradient dependent perturbation x ↦ f ( z , x , y ) , we do not impose any bilateral growth condition.

Let $\Omega\subset \mathbb{C}^{n}$ be a bounded $m$-hyperconvex domain, where $m$ is an integer such that $1\leq m\leq n$. Let $\mu$ be a positive Borel measure on $\Omega$. We show that if the complex Hessian equation $H_m (u) = \mu$ admits a (weak) subsolution in $\Omega$, then it admits a (weak) solution with a prescribed least maximal $m$-subharmonic majorant in $\Omega$.

In this paper, we establish some Schwarz type lemmas for mappings $$\Phi $$ satisfying the inhomogeneous biharmonic Dirichlet problem $$ \Delta (\Delta (\Phi )) = g$$ in $${\mathbb D}$$ , $$\Phi =f$$ on $${\mathbb T}$$ and $$\partial _n \Phi =h$$ on $${\mathbb T}$$ , where g is a continuous function on $$\overline{{\mathbb D}}$$ , f, h are continuous functions on $${\mathbb T}$$ , where $${\mathbb D}$$ is the unit disc of the complex plane $${\mathbb C}$$ and $${\mathbb T}=\partial {\mathbb D}$$ is the unit circle. To reach our aim, we start by investigating some properties of generalized harmonic functions called $$T_\alpha $$ -harmonic functions. Finally, we prove a Landau-type theorem for this class of functions, when $$\alpha >0$$ .

We prove the existence, uniqueness, and stability of a solution to the Dirichlet problem in a domain Ω ⊂ ℝn, n ≥ 1, for a nonlinear elliptic operator with potentials in the space of multiplicators, where Ω is an unbounded domain or coincides with ℝn.

We use the concept of the inside-(a, η, h) domain to construct a subsolution to the Dirichlet problem for minimal graphs over convex domains in hyperbolic space. As an application, we prove that the Holder exponent max{1/a, 1/(n + 1)} for the problem is optimal for any a ∈ [2, +∞].

We give a systematic treatment of caloric measure null sets on the essential boundary $\partial_eE$ of an arbitrary open set $E$ in ${\bf R}$. We discuss two characterisations of such sets and present some basic properties. We investigate the dependence of caloric measure null sets on the open set $E$. Thus, if $D$ is an open subset of $E$ and $Z\subseteq\partial_eE\cap\partial_eD$, we show that $Z$ is caloric measure null for $D$ if it is caloric measure null for $E$. We also give conditions on $E$ and $Z$ which imply that the reverse implication is true. We know from \cite{watson2011} that any polar subset of $\partial_eD$ is caloric measure null for $D$, but the reverse implication is not generally true. In our final result we show that, for subsets of a certain component of $\partial_eD$, caloric measure null sets are necessarily polar.

In this paper, we introduce a Finite Difference Fictitious Domain Wavelet Method (FDFDWM) for solving two dimensional (2D) linear elliptic partial differential equations (PDEs) with Dirichlet boundary conditions on regular geometric domain. The method reduces the 2D PDE into a 1D system of ordinary differential equations and applies a compactly supported wavelet to approximate the solution. The problem is embedded in a fictitious domain to aid the enforcement of the Dirichlet boundary conditions. We present numerical analysis and show that our method yields better approximation to the solution of the Dirichlet problem than traditional methods like the finite element and finite difference methods.

The definition of a Green’s function of a Cauchy–Dirichlet problem for the hyperbolic equation in a quarter plane is given. Its existence and uniqueness have been proven. Representation of the Green’s function is given. It is shown that the Green’s function can be represented by the Riemann–Green function.

This paper deals with boundary value problems for local and nonlocal Laplace operator in 2D with exponential nonlinearities, the so-called Liouville type equations. They include the mean field equation and other equations arising in the statistical mechanics. Existence results into an explicit form for the Dirichlet problem in the unit disc $B_{1} \subset {\mathbf{R}}^{2} $ B 1 ⊂ R 2 and in the participation of positive parameters in the right-hand sides are proved in Theorems 2 and 3. Theorem 2 is illustrated by several examples including an application to the differential geometry. In Theorem 4 global radial solution of the Cauchy problem with constant data at $\partial B_{1} $ ∂ B 1 and under appropriate conditions is constructed. It develops logarithmic singularities for $r = 0 $ r = 0 , $r = \infty $ r = ∞ . An illustrative example to Theorem 4 in the case of two exponents is given at the end of the paper.

In this paper, we consider the Dirichlet problem for harmonic functions on a two-dimensional complex of a special type. We prove that this problem is a Fredholm problem in the Holder class and its index is zero.

The study of the Dirichlet problem in the unit disk 𝔻 with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua [48] has been devoted to boundary-value problems (only with Holder continuous data) for the generalized analytic functions, i.e., continuous complex valued functions h(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form 𝜕zh + ah + b $$ \overline{h} $$ = c ; where it was assumed that the complex valued functions a; b and c belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar´e and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called A−harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions h : D → ℂ with the sources g : 𝜕zh = g ∈ Lp, p > 2 , and to generalized harmonic functions U with sources G : △U = G ∈ Lp, p > 2. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar´e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.

In this paper we establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary. This expansion formula shows the singularity profile of solutions at the boundary. We deal with both linear and nonlinear elliptic equations, including fully nonlinear elliptic equations and equations of Monge-Ampere type.

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction which is dependent on the gradient. We look for positive solutions and we do not assume that the reaction is nonnegative. Using a mixture of variational and topological methods (the "frozen variable" technique), we prove the existence of a positive smooth solution.

We study the Dirichlet problem in a half-space for elliptic differential-difference equations with operators that are compositions of differential operators and shift operators not bound by commensurability conditions for shifts. For this problem, we establish classical solvability or solvability almost everywhere (depending on the constraints imposed on the boundary data), construct an integral representation of the solution by means of a Poisson-type formula, and prove that it approaches zero as the time-like independent variable tends to infinity.

For a partial differential equation of order 2n with constant coefficients in the domain G := {x = (x1,…, xm) : 0 < xj < 1 < ∞, j = 1,…, m, m ϵ ℕ} , we study the problem with conditions that are multipoint perturbations of the Dirichlet boundary conditions by using the Fourier method. To investigate the spectral properties of a multipoint problem, we use the operator of transformation R: L2 (G) → L2 (G) that establishes the relationship RL0 = LR between the self-adjoint operator L0 of the Dirichlet problem and the operator L of multipoint problem. The solution of the problem with homogeneous multipoint conditions is constructed in the form of Fourier series in the system of eigenfunctions of the operator of the problem. Moreover, the conditions for its existence and uniqueness are established.

In this paper, the author solves the Dirichlet problem for Hermitian-Poisson metric equation $$\sqrt { - 1} {\Lambda _\omega }{G_H} = \lambda {\rm{Id}}$$ and proves the existence of Hermitian-Poisson metrics on flat bundles over a class of complete Hermitian manifolds. When λ = 0, the Hermitian-Poisson metric is a Hermitian harmonic metric.

At present, the fundamental solutions of the multidimensional singular elliptic equation are known and they are expressed in terms of the well-known Lauricella hypergeometric function of several variables. In this paper, we study the Dirichlet problem for an elliptic equation with several singular coefficients in an unbounded domain. When finding the solution to the posed problem, the expansion and summation formulas are used, as well as the limit relation for the Lauricella hypergeometric function of many variables.

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a Caratheodory reaction f(z, x) which is ( $$p-1$$ )-superlinear in $$x\in {\mathbb {R}}$$ but without satisfying the Ambrosetti–Rabinowitz condition. Using variational tools and critical groups, we prove a multiplicity theorem producing three nontrivial smooth solutions all with sign information (one positive, one negative and one nodal).

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator. The reaction has a parametric concave term and negative sublinear perturbation. In contrast to the case of a positive perturbation, we show that now for all big values of the parameter $$\lambda >0$$ , we have at least two positive solutions which do not vanish in the domain. In the process we prove a nonlinear maximum principle which is of independent interest.

In this paper we consider a Dirichlet problem driven by an anisotropic $(p,q)$-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map., Comment: arXiv admin note: text overlap with arXiv:2007.00945

A curvilinear integral equation of the exterior Dirichlet boundary value problem for the Laplace equation is considered. A new method is proposed to construct a quadrature formula for a singular integral. The method is used to derive a quadrature formula for the normal derivative of the double layer logarithmic potential. For specifically chosen control points, the equation is replaced by a system of algebraic equations, and the existence and uniqueness of a solution of this system are established. The convergence of the solution of this system to the exact solution of the integral equation is proved, and the rate of convergence of the method is deduced.