Dirichlet Problem Research Articles

On the Dirichlet problem for the Schrödinger equation in the upper half-space

On the Dirichlet problem for the Schrödinger equation in the upper half-space

Positive solutions for singular problems with multivalued convection

We consider an elliptic Dirichlet problem, driven by a general nonlinear, nonhomogeneous differential operator and a reaction that has the competing effects of a parametric singular term and of a multivalued perturbation which is gradient dependent (convection). Using truncations, comparisons and the theory of nonlinear operators of monotone type, we show that for all small values of the parameter λ>0, the problem has a positive and smooth solution.

Solution to the Dirichlet Problem of the Wave Equation on a Star Graph

In this paper, the solution to the Dirichlet problem for the wave equation on the star graph is constructed. To begin, we solve the boundary value problem on the interval (on one edge of the graph). We use the generalized functions method to obtain the wave equation with a singular right-hand side. The solution to the Dirichlet problem is determined through the convolution of the fundamental solution with the singular right-hand side of the wave equation. Thus, the solution found on the interval is determined by the initial functions, boundary functions, and their derivatives (the unknown boundary functions). A resolving system of two linear algebraic equations in the space of the Fourier transform in time is constructed to determine the unknown boundary functions. Following inverse Fourier transforms, the solution to the Dirichlet problem of the wave equation on the interval is constructed. After determining all the solutions on all edges and taking the continuity condition and Kirchhoff joint condition into account, we obtain the solution to the wave equation on the star graph.

Dirichlet Problem on the Half-Line for an Abstract Euler–Poisson–Darboux Equation Containing Powers of an Unbounded Operator

Dirichlet Problem on the Half-Line for an Abstract Euler–Poisson–Darboux Equation Containing Powers of an Unbounded Operator

Asymptotics of the Solution of the Bisingular Dirichlet Problem for a Ring with an Intermediate Boundary Layer

Asymptotics of the Solution of the Bisingular Dirichlet Problem for a Ring with an Intermediate Boundary Layer

On the Estimate of Integral Norm of Solution of the Dirichlet Problem for Quasilinear Elliptic Equation

On the Estimate of Integral Norm of Solution of the Dirichlet Problem for Quasilinear Elliptic Equation

A Decomposition for Borel Measures \(\mu \le \mathcal{H}^{s}\)

We prove that every finite Borel measure \( \mu \) in \( \mathbb R^N \) that is bounded from above by the Hausdorff measure \( \mathcal{H}^s \) can be split in countable many parts \( \mu{\lfloor_{_{{E_k}}}} \) that are bounded from above by the Hausdorff content \( \mathcal{H}^s_\infty \). Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to give a simpler proof for the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.

On variational competing [formula omitted]Laplacian Dirichlet problem with gradient depending weight

We investigate the existence of variational generalized solutions to coercive competitive equations driven by a competing p,q−Laplacian type operator with weigth depending on the gradient. An abstract existence result based on a Galerkin scheme is therefore obtained and subsequently applied.

Equilibrium Configurations for Nonhomogeneous Linearly Elastic Materials with Surface Discontinuities

We prove a compactness and semicontinuity result that applies to minimisation problems in nonhomogeneous linear elasticity under Dirichlet boundary conditions. This generalises a previous compactness theorem that we proved and employed to show existence of minimisers for the Dirichlet problem for the (homogeneous) Griffith energy.

Infinitely Many Solutions for a Perturbed Partial Discrete Dirichlet Problem Involving ϕc-Laplacian

In this paper, by using critical point theory, the existence of infinitely many small solutions for a perturbed partial discrete Dirichlet problems including the mean curvature operator is investigated. Moreover, the present study first attempts to address discrete Dirichlet problems with ϕc-Laplacian operator in relative to some relative existing references. Based on our knowledge, this is the research of perturbed partial discrete bvp with ϕc-Laplacian operator for the first time. At last, two examples are used to examplify the results.

A comparison principle for doubly nonlinear parabolic partial differential equations

In this paper, we derive a comparison principle for non-negative weak sub- and super-solutions to doubly nonlinear parabolic partial differential equations whose prototype is ∂tuq-div(|∇u|p-2∇u)=0inΩT,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t u^q - {{\\,\ extrm{div}\\,}}{\\big (|\ abla u|^{p-2}\ abla u \\big )}=0 \\qquad \ ext{ in } \\Omega _T, \\end{aligned}$$\\end{document}with q>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q>0$$\\end{document} and p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document} and ΩT:=Ω×(0,T)⊂Rn+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _T:=\\Omega \ imes (0,T)\\subset \\mathbb {R}^{n+1}$$\\end{document}. Instead of requiring a lower bound for the sub- or super-solutions in the whole domain ΩT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _T$$\\end{document}, we only assume the lateral boundary data to be strictly positive. The main results yield some applications. Firstly, we obtain uniqueness of non-negative weak solutions to the associated Cauchy–Dirichlet problem. Secondly, we prove that any weak solution is also a viscosity solution.

Interface potential in composites with general imperfect transmission conditions

The model analyzed in this paper has its origins in the description of composites made by a hosting medium containing a periodic array of inclusions coated by a thin layer consisting of sublayers of two different materials. This two-phase coating material is such that the external part has a low diffusivity in the orthogonal direction, while the internal one has high diffusivity along the tangential direction. In a previous paper (Amar in IFB 21:41–59, 2019), by means of a concentration procedure, the internal layer was replaced by an imperfect interface. The present paper is concerned with the concentration of the external coating layer and the homogenization, via the periodic unfolding method, of the resulting model, which is far from being a standard one. Despite the fact that the limit problem looks like a classical Dirichlet problem for an elliptic equation, in the construction of the homogenized matrix and of the source term, a very delicate analysis is required.

Existence and non-existence of minimal graphs

We study the Dirichlet problem for minimal surface systems in arbitrary dimension and codimension via mean curvature flow, and obtain the existence of minimal graphs over arbitrary mean convex bounded C2 domains for a large class of prescribed boundary data. This result can be seen as a natural generalization of the classical sharp criterion for solvability of the minimal surface equation by Jenkins-Serrin. In contrast, we also construct a class of prescribed boundary data on just mean convex domains for which the Dirichlet problem in codimension 2 is not solvable. Moreover, we study existence and the uniqueness of minimal graphs by perturbation.

Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in $${\text {CAT}}(1)$$ spaces

Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in $${\text {CAT}}(1)$$ spaces

On approximation of the Dirichlet problem for divergence form operators by Robin problems

We show that, under natural assumptions, solutions of Dirichlet problems for uniformly elliptic divergence form operators can be approximated pointwise by solutions of some versions of Robin problems. The proof is based on the stochastic representation of solutions and properties of reflected diffusions corresponding to divergence form operators.

Homogenization of boundary value problems in perforated Lipschitz domains

This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet, regularity, and Neumann problems with L2 boundary data in a periodically perforated Lipschitz domain.

Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves

Abstract We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Padé approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition). We prove upper- and lower-bounds on the relative error incurred by this approximation, both in the whole domain and in a fixed neighbourhood of the obstacle (i.e., away from the artificial boundary). Our bounds are valid for arbitrarily-high frequency, with the artificial boundary fixed, and show that the relative error is bounded away from zero, independent of the frequency, and regardless of the geometry of the artificial boundary.

Solving a Dirichlet problem on unbounded domains via a conformal transformation

In this paper, we solve the p-Dirichlet problem for Besov boundary data on unbounded uniform domains with bounded boundaries when the domain is equipped with a doubling measure satisfying a Poincaré inequality. This is accomplished by studying a class of transformations that have been recently shown to render the domain bounded while maintaining uniformity. These transformations conformally deform the metric and measure in a way that depends on the distance to the boundary of the domain and, for the measure, a parameter p. We show that the transformed measure is doubling and the transformed domain supports a Poincaré inequality. This allows us to transfer known results for bounded uniform domains to unbounded ones, including trace results and Adams-type inequalities, culminating in a solution to the Dirichlet problem for boundary data in a Besov class.

On the Dirichlet problem for A-harmonic functions

We study the Dirichlet boundary value problem with continuous boundary data for the A-harmonic equations div[A grad u] = 0 in an arbitrary bounded domain D of the complex plane £ with no boundary component degenerated to a single point. We provide integral criteria, including the BMO and FMO criteria expressed in terms of A (z), for the existence of weak solutions to the problem. We also discuss the connections between A-harmonic functions and potential theory.

A time‐domain boundary element method for the 3D dissipative wave equation: Case of Neumann problems

AbstractThe present article proposes a time‐domain boundary element method (TDBEM) for the three‐dimensional (3D) dissipative wave equation (DWE). Although the fundamental ingredients such as the Green's function for the 3D DWE have been known for a long time, the details of formulation and implementation for such a 3D TDBEM have been unreported yet to the author's best knowledge. The present formulation is performed truly in time domain on the basis of the time‐dependent Green's function and results in a marching‐on‐in‐time fashion. The main concern is in the evaluation of the boundary integrals. For this regard, weakly‐ and removable‐singularities are carefully treated. The proposed TDBEM is checked through the numerical examples whose solutions can be obtained semi‐analytically by means of the inverse Laplace transform. The results of the present TDBEM are satisfactory to validate its formulation and implementation for Neumann problems. On the other hand, the present formulation based on the ordinary boundary integral equation (BIE) is unstable for Dirichlet problems. The numerical analyses for the non‐dissipative case imply that the instability issue can be partially resolved by using the Burton–Miller BIE even in the dissipative case.