Abstract We study the Dirichlet problem for the complex Monge–Ampère equation on a strictly pseudoconvex domain in or a Hermitian manifold. Under the condition that the right‐hand side lies in an function and the boundary data are Hölder continuous, we prove the global Hölder continuity of the solution.

Abstract In this paper, we consider the following Dirichlet problem for the fully nonlinear elliptic equation of Grad-Mercier type under asymptotic convexity conditions { F ( D 2 u ( x ) , D u ( x ) , u ( x ) , x ) = g ( E u ( x ) ) + f ( x ) in Ω , u = ψ on ∂ Ω , E u ( x ) = L d ( { y ∈ Ω : u ( y ) ⩾ u ( x ) } ) in Ω . In order to overcome the non-convexity of the operator F and the nonlocality of the nonhomogeneous term g ( E u ) , we apply the compactness methods and frozen technique to prove the existence of the L p -viscosity solutions and the global W 2 , p estimate. As an application, we derive a Cordes–Nirenberg type estimate up to boundary. Furthermore, we establish a global BMO estimate for the second derivatives of solutions by using an asymptotic approach, thereby refining the borderline case of Calderón–Zygmund estimates.

The Euclidean path integral for gravity is enriched by the addition of boundaries, which provide useful probes of thermodynamic properties. Common boundary conditions include Dirichlet conditions on the boundary induced metric; microcanonical conditions, which refers to fixing some components of the Brown-York boundary stress tensor; and conformal conditions, in which the conformal structure of the induced metric and the trace of the extrinsic curvature are fixed. Boundaries also present interesting problems of consistency. The Dirichlet problem is known, under various (and generally different) conditions, to be inconsistent with perturbative quantization of graviton fluctuations, to exhibit thermodynamic instability, or to require infinite fine-tuning in the presence of matter fluctuations. We extend some of these results to other boundary conditions. We find that similarly to the Dirichlet problem, the graviton fluctuation operator is not elliptic with microcanonical boundaries, and the nonelliptic modes correspond to “boundary-moving diffeomorphisms.” However, we argue that microcanonical factorization of path integrals—essentially, the insertion of microcanonical constraints on two-sided surfaces in the bulk—is not affected by the same issues of ellipticity. We also show that for a variety of matter field boundary conditions, matter fluctuations renormalize the gravitational bulk and boundary terms differently, so that the classical microcanonical or conformal variational problems are not preserved unless an infinite fine-tuning is performed.

ABSTRACT A generalized Fourier series method is constructed to approximate the solution of the Dirichlet problem in a finite domain with finitely many holes, in the case of bending of elastic plates with transverse shear deformation. The theoretical results are illustrated by an example with specific boundary conditions.

The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains

Polyharmonic Dirichlet problem for the high-order Poisson equation in a sector domain

The Dirichlet Problem for Nonsymmetric Augmented Hessian Quotient Type Equations

With the help of a new Picone-type identity, we prove that positive solutions u\,{\in}\, H^{s}(\mathbb{R}^{N}) to the equation (-\Delta )^{s} u+ u=u^{p} in \mathbb{R}^{N} are nonradially nondegenerate, for all s\in (0,1) , N\geq 1 and p>1 strictly smaller than the critical Sobolev exponent. By this we mean that the linearized equation (-\Delta )^{s} w+ w-pu^{p-1}w = 0 does not admit nonradial solutions besides the directional derivatives of u . Letting B be the unit centered ball and \lambda_{1}(B) the first Dirichlet eigenvalue of the fractional Laplacian (-\Delta )^{s} , we also prove that positive solutions to (-\Delta )^{s} u+\lambda u=u^{p} in {B} , with u=0 on \mathbb{R}^{N}\setminus B , are nonradially nondegenerate for any \lambda> -\lambda_{1}(B) in the sense that the linearized equation does not admit nonradial solutions. From these results, we then deduce uniqueness and full nondegeneracy of positive solutions in some special cases. In particular, in the case N=1 , we prove that the equation (-\Delta )^{s} u+ u=u^{2} in \mathbb{R} or in B , with zero exterior data, admits a unique even solution which is fully nondegenerate in the optimal range s \in (\frac{1}{6},1) , thus extending the classical uniqueness result of Amick and Toland on the Benjamin–Ono equation. Moreover, in the case N=1 , \lambda=0 , we also prove the uniqueness and full nondegeneracy of positive solutions for the Dirichlet problem in B with arbitrary subcritical exponent p . Finally, we determine the unique positive ground state solution of (-\Delta )^{\frac{1}{2}}u+ u=u^{p} in \mathbb{R}^{N} , N \ge 1 with p=1+\frac{2}{N+1} and compute the sharp constant in the associated Gagliardo–Nirenberg inequality \|u\|_{L^{p+1}(\mathbb{R}^N)}\le \rule{0pt}{9pt}C \|(-\Delta )^{\frac{1}{4}} u\|_{L^2(\mathbb{R}^N)}^{\frac{N}{N+2} \|u\|_{L^2(\mathbb{R}^N)}^{\frac{2}{N+2}}} .

In this paper we investigate the boundedness properties of generalized fractional integral on generalized weighted Morrey spaces over metric measure spaces. The measure used in this paper is a doubling measure which satisfies the growth condition. The results show that the generalized fractional integral is bounded from one generalized weighted Morrey spaces to another generalized weighted Morrey space over metric measure spaces either with the same or with the different parameters. Our results extend the known results for fractional integrals on generalized Morrey spaces. We then investigate the regularity of the solution of Dirichlet problem with the data in generalized weighted Morrey spaces by using the boundedness properties of the generalized fractional integral on generalized weighted Morrey space.

It is proven that conformal mapping is effective for constructing harmonic Green functions and harmonic Neumann functions in lens domains and circular triangle domains with the specific angle π / n , where n ≥ 2 , n ∈ Z on C . The Dirichlet and Neumann boundary problems of the Poisson equation in these domains are solved by using explicit expressions of the harmonic Green and Neumann functions, and then applying the corresponding representation formulas. Additionally, the explicit expressions of Neumann and Robin functions are modified.

This work presents an analytical solution for the steady laminar wake generated by a finite wall segment acting as a sink for heat or mass transfer. The classical Lévêque solution is extended to include the wake region downstream of the active surface by employing Laplace transform methods to couple Dirichlet and Neumann boundary value problems through convolution identities. This yields a unified closed-form expression for the scalar field that reduces to the Lévêque result above the sink and provides a new analytical expression for the wake region. Numerical simulations confirm the analytical solution, with errors decreasing systematically under mesh refinement. The derived expressions enable direct calculation of scalar recovery at any point in the wake, providing essential information for designing segmented systems where wake interference between adjacent active elements must be predicted. The solution also serves as a benchmark for numerical methods solving mixed boundary value problems in convective transport.

In this paper, we study two-dimensional Dirichlet problems (linear and nonlinear) with discontinuous coefficients, order one terms and data in L 1 (and no more).

In this note, we present a logarithmic-type upper bound for weak subsolutions to a class of integro-differential problems, whose prototype is the Dirichlet problem for the fractional Laplacian. The bound is slightly smaller than the classical one in this field.

For the numerical solution of the Dirichlet problem for the Poisson equation, both direct and iterative methods have been developed. However, the required number of arithmetic operations for direct methods, as well as the number of iterations in iterative methods, often turns out to be very large. For this reason, the issues of high accuracy and efficiency of various methods remain relevant. In this work, a new high-accuracy and efficient method is proposed for the numerical solution of the Dirichlet problem for the Poisson equation – a discrete version of the preliminary integration method, which significantly surpasses existing direct and iterative methods in terms of the number of arithmetic operations. The efficiency of the proposed method is illustrated by tabular and graphical results.

We consider a Dirichlet problem driven by a differential operator with unbalanced growth and a reaction exhibiting the combined effects of a parametric singular term and a resonant perturbation. Using a combination of variational tools and critical groups, we show that, for all small values of the parameter, the problem has at least two bounded positive solutions.

We investigate the structural stability of solutions to boundary value problems for the variable exponent $ p(x) $-Laplacian. Stability questions for such problems under perturbations of the boundary operator, the differential operator, boundary data, or the domain have a long history and play a central role in the analysis of nonlinear partial differential equations (PDEs). In this work, we consider the Poisson boundary value problem with nonhomogeneous boundary conditions and study the behavior of its solutions under variations of the exponent functions $ p(x) $. Our results extend the classical stability theorem of Lindqvist (1987), originally formulated for constant $ p $, to the variable exponent setting. Moreover, our approach sharpens and generalizes the framework developed by Zhikov (2011), allowing for nonhomogeneous boundary data and providing stronger convergence results for the associated family of solutions. Specifically, it is shown that if the sequence $ (p_j(x)) $ increases uniformly to $ p(x) $ in a bounded, smooth domain $ \Omega $, then the sequence $ (u_i) $ of solutions to the Dirichlet problem for the $ p_i(x) $-Laplacian with fixed boundary datum $ \varphi $ converges (in a sense to be made precise) to the solution $ u_p $ of the Dirichlet problem for the $ p(x) $-Laplacian with boundary datum $ \varphi $. A similar result is proved for a decreasing sequence $ p_j\searrow p $.

Dirichlet problem for diffusions with jumps

In this paper, certain linear operators with argument transformations are defined within the class of smooth functions. These operators are introduced using matrices of involution-type mappings. Subsequently, the specified operators are used to define a non-local analogue of the Laplace operator and corresponding boundary operators. For the obtained non-local analogue of the Poisson equation, solvability questions for certain boundary value problems are investigated. The boundary conditions of the considered problems are specified as relations between the values of the unknown function at different points and, thus, belong to Bitsadze–Samarskii type problems. Theorems on the existence and uniqueness of the solution to the investigated problems are proved. It is shown that the well-posedness of the considered problems depends essentially on the coefficients of the introduced linear transformation operators. Using the Green's function for the classical Dirichlet and Neumann problems, the explicit form of the Green's function for the considered problems is constructed. Moreover, integral representations of the solutions to these problems are also obtained using the constructed Green's function. In addition, the study examines the structure of the transformation operators and analyzes their properties that influence the stability of the solution. The obtained results are compared with classical local models, which makes it possible to identify the advantages of the nonlocal approach. It is noted that the proposed methods can also be applied to other types of elliptic equations involving argument transformations.

This article presents a study on the analytical solution of the Dirichlet problem for the Laplace equation in two-dimensional space. The primary focus is on the Green's function, which is a key tool for solving such problems. We considered cases where the sources are located on geometrically simple sets: on a straight line, a unit circle, and a line segment.We developed and applied an effective method for calculating the sum of harmonic functions $h_k(z,a_k)$, which are the correcting terms in the Green's function expression. This allowed us to obtain analytical formulas for the potential generated by a system of point sources located in the specified configurations. Specifically, for each case, we found an estimate for the total potential.The findings are of significant value to theoretical physics and engineering applications, particularly in electrostatics, heat conduction, and hydrodynamics, where similar boundary value problems arise. The proposed approach can serve as a basis for further research aimed at solving more complex problems with sources located on curved or higher-dimensional manifolds.

The purpose of this paper is to give sufficient conditions for the existence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain \(\Omega\) in \(R^{n}\). Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are super-sub solutions method, eigenvalues of operators, maximum principles, spectrum estimates, inverse function theory, and general elliptic theory. The arguments also rely on some detailed properties for the solution of logistic equations. These results yield an algebraically computable criterion for the positive coexistence of species of animals with predator-prey relation in many biological models.