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This paper discusses the first exit and Dirichlet problems of the nonisotropic tempered α -stable process X t . The upper bounds of all moments of the first exit position X τ D and the first exit time τ D are explicitly obtained. It is found that the probability density function of X τ D or τ D exponentially decays with the increase of X τ D or τ D , and E τ D ∼ E X τ D - E X τ D 2 , E τ D ∼ E X τ D . Next, we obtain the Feynman–Kac representation of the Dirichlet problem by employing the semigroup theory. Furthermore, averaging the generated trajectories of the stochastic process leads to the solution of the Dirichlet problem, which is also verified by numerical experiments. [ABSTRACT FROM AUTHOR]

We establish Lyapunov‐type inequality for equations concerning general class of second‐order non‐symmetric elliptic operators with singular coefficients. Our approach is based on the probabilistic representation of solutions and stochastic calculus. We also discuss a Lyapunov‐type inequality for equations pertaining to second‐order symmetric operator with some regularity assumptions on the coefficients and a nonlinear Neumann boundary condition. [ABSTRACT FROM AUTHOR]

Let Ω⊂Rn$$ \Omega \subset {R}^n $$ be a bounded domain with sufficiently smooth boundary ∂Ω$$ \mathrm{\partial \Omega } $$ and XΩ$$ X\left(\Omega \right) $$ be a symmetric space defined on the measure space Ω;dx$$ \left(\Omega; dx\right) $$. We consider a Dirichlet problem for 2m$$ 2m $$‐th order polyharmonic equation, and we establish its solvability (in strong sense) in Sobolev space WX2mΩ$$ {W}_X^{2m}\left(\Omega \right) $$ generated by the norm of XΩ$$ X\left(\Omega \right) $$. Such spaces include classical Sobolev spaces, Orlicz–Sobolev spaces, grand Sobolev spaces, and Marcinkiewicz–Sobolev spaces. The obtained results are new for these special cases, too. [ABSTRACT FROM AUTHOR]

In this paper, we explore the ball convergence properties of enhanced Phragmén–Lindelöf type methods for solving the Dirichlet problem with an elliptic operator. By placing requirements on the elliptic operator and auxiliary points, we characterize the ball limit of a series of elliptic operators in non‐divergence form. Considering all dimensions, it becomes apparent that dealing solely with measurable and bounded coefficients for this class of operators is insufficient, necessitating additional regularity assumptions on them. We find the radii of convergence balls using recurrence relations, so as to guarantee the convergence of iterative reconstruction techniques, beginning from any point inside the ball centered on an auxiliary point. [ABSTRACT FROM AUTHOR]

The main crux of this manuscript is to study the existence of weak solutions for the following fractional double phase elliptic problem with Dirichlet boundary conditions (- Δ) p σ w + (- Δ) q σ w = ψ (z) + g (z , w) in Θ , w = 0 , on R N \ Θ , where 1 < p < q < N σ , σ ∈ (0 , 1) and (- Δ) p σ , (- Δ) q σ are fractional Laplacian operators defined in the principle value sense. The existence results are obtained by combining the Galerkin approximation, Lebesgue and Sobolev spaces, the Young measures approach, and the assumption of growth conditions on a given datum g . Our results expand upon and broaden several recent studies in this area of literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider the BV least gradient problem with Dirichlet condition on a part Γ ⊂ ∂ ⁡ Ω {\Gamma\subset\partial\Omega} and Neumann boundary condition on its complementary part ∂ ⁡ Ω \ Γ {\partial\Omega\backslash\Gamma} . We will show that in the plane this problem is equivalent to an optimal transport problem with import/export taxes on ∂ ⁡ Ω \ Γ {\partial\Omega\backslash\Gamma} . Thanks to this equivalence, we will be able to show existence and uniqueness of a solution to this mixed least gradient problem and we will also prove some Sobolev regularity on this solution. We note that these results generalize those in [S. Dweik, W 1 , p W^{1,p} regularity on the solution of the BV least gradient problem with Dirichlet condition on a part of the boundary, Nonlinear Anal. 223 2022, Article ID 113012], where we studied the pure Dirichlet version of this problem. [ABSTRACT FROM AUTHOR]

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L 2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L 2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L -traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods. [ABSTRACT FROM AUTHOR]

The aim of this work is to present a result of multiplicity of solutions, in generalized Sobolev spaces, for a non-local elliptic problem with p(x)-Laplace operator containing k distinct critical Sobolev-Hardy exponents combined with singularity points U ... on Ω, where Ω ⊂ ℝN is a bounded domain with 0 ∈ Ω and 1 < p- ≤ p(x) ≤ p+ < N. The real function M is bounded in [0, +∞) and the functions hi (i = 1,...,k) and f are functions whose properties will be given later. To obtain the result we use the Lions' concentration-compactness principle for critical Sobolev-Hardy exponent in the space W01,p(x)(Ω) due to Yu, Fu and Li, and the Fountain Theorem. [ABSTRACT FROM AUTHOR]

We consider a nonlinear Dirichlet problem driven by the double phase operator. The reaction has the combined effects of parametric concave term and of an indefinite convex one ("concave-convex" problem with indefinite weight). Using the Nehari method we prove the existence of two bounded, positive ground state solutions. [ABSTRACT FROM AUTHOR]

In this manuscript we deal with elliptic equations with superlinear first order terms in divergence form of the following type − div (M (x) ∇ u) = − div (h (u) E (x)) + f (x) , where M is a bounded elliptic matrix, the vector field E and the function f belong to suitable Lebesgue spaces, and the function s → h (s) features a superlinear growth at infinity. We provide some existence and non existence results for solutions to the associated Dirichlet problem and a comparison principle. [ABSTRACT FROM AUTHOR]

We consider a model initial‐ and Dirichlet boundary–value problem for a nonlinear Schrödinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second‐order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal‐order error estimates in the L2$L^2$ and the H1$H^1$ norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a Courant‐Friedrichs‐Lewy (CFL) condition between the space mesh size and the time step sizes. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet problem for $p(x)$ -Laplacian equations of the form $$ \begin{align*} -\Delta_{p(x)}u+b(x)\vert u\vert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega). \end{align*} $$ The odd nonlinearity $f(x,u)$ is $p(x)$ -sublinear at $u=0$ but the related limit need not be uniform for $x\in \Omega $. Except being subcritical, no additional assumption is imposed on $f(x,u)$ for $|u|$ large. By applying Clark's theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function $u=0$. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of weak solutions of the p (x) -Laplacian problems that are subject to Dirichlet boundary conditions, employing variational techniques as our primary analytical framework. [ABSTRACT FROM AUTHOR]

Non-axisymmetric frictionless JKR-type adhesive contact between a rigid body and a thin incompressible elastic layer bonded to a rigid base is considered in the framework of the leading-order asymptotic model, which has the form of an overdetermined boundary value problem. Based on the first-order perturbation of the Neumann operator in the Dirichlet problem for Poisson's equation, the decohesion initiation problem is formulated in the form of a variational inequality. The asymptotic model assumes that the contact zone and its boundary contour during the detachment process are unknown. The absence of the solvability theorem is illustrated by an example of the instability of an axisymmetric flat circular contact. [ABSTRACT FROM AUTHOR]

In this paper, the existence of weak solutions to some Dirichlet problems with fractional competing operators and distributional Riesz fractional gradient is investigated. Due to the nature of driving operators, the most known techniques, basically based on ellipticity and monotonicity, are no longer applicable. Generalized solutions (in a suitable sense) are obtained via an approximation procedure and a corollary of the Brouwer fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper we study the regularity and the behavior in time of the solutions to the Dirichlet problem for the total variation flow in a bounded domain. We show that the regularity of the initial datum influences the behavior and can produce extinction in finite time and “immediately boundedness of the solutions”. [ABSTRACT FROM AUTHOR]

We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated with fully nonlinear elliptic equations of order $2\,s$ , with $s\in (1/2,\,1)$ , and a coercive gradient term with subcritical power $0. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0 , involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry–Lions for the subquadratic case $1. [ABSTRACT FROM AUTHOR]

In the field of enumeration of walks in cones, it is known how to compute asymptotically the number of excursions (finite paths in the cone with fixed length, starting and ending points, using jumps from a given step set). As it turns out, the associated critical exponent is related to the eigenvalues of a certain Dirichlet problem on a spherical domain. An important underlying question is to decide whether this asymptotic exponent is a (non-)rational number, as this has important consequences on the algebraic nature of the associated generating function. In this paper, we ask whether such an excursion sequence might admit an asymptotic expansion with a first rational exponent and a second non-rational exponent. While the current state of the art does not give any access to such many-term expansions, we look at the associated continuous problem, involving Brownian motion in cones. Our main result is to prove that in dimension three there exists a cone such that the heat kernel (the continuous analog of the excursion sequence) has the desired rational/non-rational asymptotic property. Our techniques come from spectral theory and perturbation theory. More specifically, our main tool is a new Hadamard formula, which has an independent interest and allows us to compute the derivative of eigenvalues of spherical triangles along infinitesimal variations of the angles. [ABSTRACT FROM AUTHOR]

In this work, we shall be interested in the existence of a solution to the following Dirichlet problem for a specific class of elliptical anisotropic equations of the type ... where Ω is a bounded open set of ℝN, A = Σ|α|∞=0(- 1)|α|Dα(αα|Dαu|Pα(x)-2Dαu) is an operator of infinite order and g(x, s) is a non-linear lower order term that verify some natural growth and sign conditions, where the data f is framed in L¹(Q). [ABSTRACT FROM AUTHOR]

We consider the boundary problem (1) − (r (x) y ′ (x)) ′ + q (x) y (x) = f (x) , x ∈ R , (2) lim | x | → ∞ ⁡ y (x) = 0 under the following conditions: 1) r > 0 , 1 r ∈ L 1 loc (R) , q ∈ L 1 loc (R) ; 2) equation (1) is correctly solvable in L p (R) , p ∈ (1 , ∞). We obtain necessary and sufficient requirements for the functions r and q under which, regardless of the choice of a function f ∈ L p (R) , p ∈ (1 , ∞) , the solution y ∈ L p (R) of equation (1) satisfies (2). [ABSTRACT FROM AUTHOR]

The Monge-Ampère type equations over bounded convex domains arise in a host of geometric applications. In this paper, we focus on the Dirichlet problem for a class of Monge-Ampère type equations, which can be degenerate or singular near the boundary of convex domains. Viscosity subsolutions and viscosity supersolutions to the problem can be constructed via comparison principle. Finally, we demonstrate the existence, uniqueness and a series of interior regularities (including W 2 , p with p ∈ (1 , + ∞) , C 1 , μ with μ ∈ (0 , 1) , and C ∞) of the viscosity solution to the problem. [ABSTRACT FROM AUTHOR]

We study the optimal boundary regularity of solutions to Dirichlet problems involving the logarithmic Laplacian. Our proofs are based on the construction of suitable barriers via the Kelvin transform and direct computations. As applications of our results, we show a Hopf-type lemma for nonnegative weak solutions and the uniqueness of solutions to some nonlinear problems. [ABSTRACT FROM AUTHOR]

The main focus of this paper lies in investigating the existence of infinitely many positive weak solutions for the following elliptic-Kirchhoff equation with Dirichlet boundary condition The methodology adopted revolves around the technical approach utilizing the direct variational method within the framework of anisotropic variable exponent Sobolev spaces. [ABSTRACT FROM AUTHOR]

Let D = {{ C : ||| < 1} be the open unit disk in the complex plane C and let H(D) be the space of all holomorphic functions on D. For a non-negative integer n and a function f H(D), the nthh order differentiation operator is defined as Dnf = f(n). The weighted composition operator together with nthh order differentiation operator give rise to a new operator generally termed as generalized weighted composition operator denoted by Wn, and is defined by Wn, f(() = (()f(n)(((()), f H(D); D, where H(D) and is a holomorphic self-map of D. This operator is basically the combination of multiplication operator MM, composition operator CC and nthh order differentiation operator Dn. We study the boundedness and compactness of this operator between Dirichlet-type spaces and Bloch-type spaces. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the solvability of a new class of nonlocal boundary value problems for the Poisson equation is studied. Nonlocal conditions are specified in the form of a connection between the values of the unknown function at different points of the boundary. In this case, the boundary operator is determined using matrices of involution-type mappings. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Using Green’s functions of the classical Dirichlet and Neumann boundary value problems, Green’s functions of the studied problems are constructed and integral representations of solutions to these problems are obtained.

We are considered with the following nonlinear Schrödinger equation: −Δu+(λa(x)+1)u=f(u),x∈V,$$ -\Delta u+\left(\lambda a(x)+1\right)u=f(u),x\in V, $$on a locally finite graph G=(V,E)$$ G=\left(V,E\right) $$, where V$$ V $$ denotes the vertex set, E$$ E $$ denotes the edge set, λ>1$$ \lambda >1 $$ is a parameter, f(s)$$ f(s) $$ is asymptotically linear with respect to s$$ s $$ at infinity, and the potential a:V→[0,+∞)$$ a:V\to \left[0,+\infty \right) $$ has a nonempty well Ω$$ \Omega $$. By using variational methods, we prove that the above problem has a ground state solution uλ$$ {u}_{\lambda } $$ for any λ>1$$ \lambda >1 $$. Moreover, we show that as λ→∞$$ \lambda \to \infty $$, the ground state solution uλ$$ {u}_{\lambda } $$ converges to a ground state solution of a Dirichlet problem defined on the potential well Ω$$ \Omega $$. [ABSTRACT FROM AUTHOR]

The Dirichlet problem of elliptic equation from two phase problems, b (u) − div (| ∇ u) | p (x) − 2 ∇ u + μ (x) | ∇ u | q (x) − 2 ∇ u) = f (x) in Ω, u = 0 on ∂Ω is considered, where Ω is a bounded domain with a smooth boundary in R N , b is a continuous and nondecreasing function. By the theory of the weighted variable Sobolev space, the existence and uniqueness of an entropy solution for L 1 -data f are proved. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of a Dirichlet boundary value problem for a system of two coupled anisotropic Darcy–Forchheimer–Brinkman equations on a bounded Lipschitz domain in ℝn(n=2,3)$$ {\mathrm{\mathbb{R}}}&#x0005E;n\left(n&#x0003D;2,3\right) $$. Using variational methods and fixed point techniques, we obtain a well‐posedness result in L2$$ {L}&#x0005E;2 $$‐based Sobolev spaces for sufficiently small data. As an application, we investigate numerically the lid‐driven flow problem in a square cavity saturated with a bidisperse porous medium and analyze the effect of various physical parameters on the fluid motion. [ABSTRACT FROM AUTHOR]

We establish the Pogorelov-type estimates for degenerate prescribed |$k$| -curvature equations as well as |$k$| -Hessian equations. Furthermore, we investigate the interior |$C^{1,1}$| regularity of the solutions for Dirichlet problems. These techniques also enable us to improve the existence theorem for an asymptotic Plateau-type problem in hyperbolic space. [ABSTRACT FROM AUTHOR]

In this article, we propose novel boundary treatment algorithms to avoid order reduction when implicit-explicit Runge-Kutta time discretization is used for solving convection-diffusion-reaction problems with time-dependent Dirichlet boundary conditions. We consider Cartesian meshes and PDEs with stiff terms coming from the diffusive parts of the PDE. The algorithms treat boundary values at the implicit-explicit internal stages in the same way as the interior points. The boundary treatment strategy is designed to work with multidimensional problems with possible nonlinear advection and source terms. The proposed methods recover the designed order of convergence by numerical verification. For the spatial discretization, in this work, we consider local discontinuous Galerkin methods, although the developed boundary treatment algorithms can operate with other discretization schemes in space, such as finite differences, finite elements, or finite volumes. [ABSTRACT FROM AUTHOR]

The classical Minkowski problem for convex bodies has deeply influenced the development of differential geometry. During the past several decades, abundant mathematical theories have been developed for studying the solutions of the Minkowski problem; however, the numerical solution of this problem has been largely left behind, with only a few methods available to achieve that goal. In this article, focusing on the two-dimensional Minkowski problem with Dirichlet boundary conditions, we introduce two solution methods, both based on operator-splitting. One of these two methods deals directly with the Dirichlet condition, while the other one uses an approximation à la Robin of this Dirichlet condition. The relaxation of the Dirichlet condition makes the second method better suited than the first one to treat those situations where the Minkowski equation (of Monge-Ampère type) and the Dirichlet condition are not compatible. Both methods are generalizations of the solution method for the canonical Monge--Ampère equation discussed by Glowinski et al. [J. Sci. Comput., 81 (2019), pp. 2271-2302]; as such they take advantage of a divergence formulation of the Minkowski problem, which makes it well suited to both a mixed finite-element approximation and the time-discretization via an operator-splitting scheme of an associated initial value problem. Our methodology can be easily implemented on convex domains of rather general shape (with curved boundaries, possibly). The numerical experiments validate both methods, showing that if one uses continuous piecewise affine finite-element approximations of the solution of the Minkowski problem and of its three second order derivatives, these two methods provide nearly second-order accuracy for the L² and L∞ norms of the approximation error, where the Minkowski--Dirichlet problem is assumed to have a smooth solution. One can easily extend the methods discussed in this article to address the solution of three-dimensional Minkowski problems. [ABSTRACT FROM AUTHOR]

We consider a Dirichlet problem driven by the anisotropic (p , q) Laplacian. In the reaction, we have a parametric partially concave term plus a "superlinear" perturbation (convex term) which need not satisfy the Ambrosetti–Rabinowitz condition. Using variational tools, we show that for all small values of the parameter λ > 0 , the problem has at least two nontrivial smooth solutions. [ABSTRACT FROM AUTHOR]

This paper aims to study the Dirichlet problem to the complex Monge–Ampère operator in pseudoconvex domains of strictly plurisubharmonic type. We prove the Hölder continuity of solutions to this problem. [ABSTRACT FROM AUTHOR]

In this paper we study the Riemann–Hilbert problem for variable exponent poly-Smirnov space on a piecewise Lyapunov boundary. At first we introduce the poly-Smirnov function space with variable exponent, we obtain a decomposition of this space and verify the existence of the angular boundary value of function in this space. Then we investigate Dirichlet problem for poly-Smirnov function space. At last we consider Riemann–Hilbert problem for variable exponent poly-Smirnov space on a piecewise Lyapunov boundary. The method is transforming this problem into independent homogeneous Dirichlet problems, and the conditions of solvability as well as the explicit solutions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate the existence of a unique weak solution for a nonlocal Kirchhoff problem under Dirichlet boundary conditions, involving the fractional φ-Laplacian operator. Our major outcome is acquired by applying variational approaches and critical points theory. In addition, we analyse the spectrum and the eigenvalues associated to this problem. At the end and under some assumptions, we give an application to the previous problems. [ABSTRACT FROM AUTHOR]

We consider two spectral problems on an equilateral rooted tree with the standard (continuity and Kirchhoff's type) conditions at the interior vertices (except of the root if it is interior) and Dirichlet conditions at the pendant vertices (except of the root if it is pendant). For the first (Neumann) problem we impose the standard conditions (if the root is an interior vertex) or Neumann condition (if the root is a pendant vertex) at the root, while for the second (Dirichlet) problem we impose the Dirichlet condition at the root. We show that for caterpillar trees the spectra of the Neumann problem and of the Dirichlet problem uniquely determine the shape of the tree. Also, we present an example of co-spectral snowflake graphs. [ABSTRACT FROM AUTHOR]

We consider a nonlinear Dirichlet problem driven by the double phase differential operator and a reaction that has the competing effects of a parametric concave term and of a convective perturbation. Using truncation and comparison techniques and the theory of nonlinear operators of monotone type, we show that for all small values of the parameter, the problem has a bounded positive solution. [ABSTRACT FROM AUTHOR]

A class of Schwarz problems with the conditions concerning the real and imaginary parts of high-order partial differentiations for polyanalytic functions was discussed first on the bicylinder. Then, with the particular solution to the Schwarz problem for polyanalytic functions, a Dirichlet problem for bi-polyanalytic functions was investigated on the bicylinder. From the perspective of series, the specific representation of the solution was obtained. In this article, a novel and effective method for solving boundary value problems, with the help of series expansion, was provided. This method can also be used to solve other types of boundary value problems or complex partial differential equation problems of other functions in high-dimensional complex spaces. [ABSTRACT FROM AUTHOR]

In this article, we consider a class of nonlinear elliptic problems, where anisotropic leading differential operator incorporates the unbounded coefficients and the nonlinear term is a convection term. We prove the solvability of degenerate Dirichlet problem with convection, i.e. the existence of at least one bounded weak solution via the theory of pseudomonotone operators, Nemytskii-type operator and a priori estimate in the degenerate anisotropic Sobolev spaces. [ABSTRACT FROM AUTHOR]

This paper deals with a time-space fractional Schrödinger equation with homogeneous Dirichlet boundary conditions. A common strategy for discretizing time-fractional operators is finite difference schemes. In these methods, the time-step size should usually be chosen sufficiently small, and subsequently, too many iterations are required which may be time-consuming. To avoid this issue, we utilize the Laplace transform method in the present work to discretize time-fractional operators. By using the Laplace transform, the equation is converted to some time-independent problems. To solve these problems, matrix transformation and improved matrix transformation techniques are used to approximate the spatial derivative terms which are defined by the spectral fractional Laplacian operator. After solving these stationary equations, the numerical inversion of the Laplace transform is used to obtain the solution of the original equation. The combination of finite difference schemes and the Laplace transform creates an efficient and easy-to-implement method for time-space fractional Schrödinger equations. Finally, some numerical experiments are presented and show the applicability and accuracy of this approach. [ABSTRACT FROM AUTHOR]

The spectral Dirichlet problem is considered in a waveguide made of a semi-infinite cylinder Π and the resonator ΘR obtained by inflating R times a fixed star-shaped domain Θ. The behavior of the scattering coefficient s(R) is studied as the parameter R grows, namely, it is verified that this coefficient moves clockwise without stops along the unit circle in the complex plane. For s(R) = -1, the proper threshold resonance occurs; it is accompanied by the appearance of an almost standing wave and provokes various near-threshold anomalies, in particular, splitting eigenvalues off from the threshold. Under the geometrical symmetry, resonances of other type are shown to be generated by trapped waves at the threshold. The justification of asymptotics is made by applying the technique of weighted spaces with detached asymptotics and an analysis of the singularities of physical fields at the edge ∂ΘR ∩ ∂Π. [ABSTRACT FROM AUTHOR]

We show that there exists only one solution (the trivial identically zero solution) for some nonlinear elliptic Dirichlet problems, involving the p-Laplacian operator and nonlinear terms with supercritical growth, in bounded contractible ramified domains, that is domains of ℝn with n ≥ 2, close to a prescribed subset of ℝn, which is contractible in itself and consists of a finite number of smooth curves. In dimension n = 2 we expect that this result might be extended to cover all the bounded contractible domains of ℝ2. On the contrary, this extension is not possible in dimension n ≥ 3 because of some counterexamples concerning existence and multiplicity of nontrivial solutions in some contractible domains that may be even arbitrarily close to starshaped domains (where the well-known Pohozaev nonexistence result holds). However, also for n ≥ 3 our result allows us to prove nonexistence of nontrivial solutions in bounded contractible domains that may be very different from the starshaped ones and even arbitrarily close to some noncontractible domains where there exist many positive and nodal solutions. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the smoothness of generalized solutions of the first boundary-value problem for a strongly elliptic functional differential equation containing orthotropic contraction transformations of the arguments of the unknown function in the leading part. The problem is considered in a disc, the coefficients of the equation are constant. Orthotropic contraction is understood as different contraction in different variables. Conditions for the conservation of smoothness on the boundaries of neighboring subdomains formed by the action of the contraction transformation group onto the disc are found in explicit form for any right-hand side from the Lebesgue space. [ABSTRACT FROM AUTHOR]

In this paper, we establish the equivalence of entropy and renormalized solutions of secondorder elliptic equations with nonlinearities defined by the Musielak–Orlicz functions and the right-hand side from the space L1(Ω). In nonreflexive Musielak–Orlicz–Sobolev spaces, we prove the existence and uniqueness of both entropy and renormalized solutions of the Dirichlet problem in domains with a Lipschitz boundary. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak strong uniqueness of the Dirichlet type problems of fractional Laplace (Poisson) equations. We construct the Green’s function and the Poisson kernel. We then provide a somewhat sharp condition for the solution to be unique. We also show that the solution under such condition exists and must be given by our Green’s function and Poisson kernel. In doing these, we establish several basic and useful properties of the Green’s function and Poisson kernel. Based on these, we obtain some further a priori estimates of the solutions. Surprisingly those estimates are quite different from the ones for the local type elliptic equations such as Laplace equations. These are basic properties to the fractional Laplace equations and can be useful in the study of related problems. [ABSTRACT FROM AUTHOR]

The present essay is concerned with providing rigorous justification of a long‐standing practice in numerical simulation of partial differential equations. Theory often sets initial‐value problems on all of R${\mathbb {R}}$ or Rd${\mathbb {R}}^d$. If the initial data are localized in space, it has been usual practice to approximate the problem by an associated periodic problem or a homogeneous Dirichlet problem set on a finite interval. While these strategies are commonplace, rigorous justification of the practice is sparse. It is our purpose here to indicate justification of this practice in the concrete context of a surface water wave model. While the theory worked out here is specific to the particular partial differential equation, it will be apparent to the reader that more general results may be derived using the same approach. [ABSTRACT FROM AUTHOR]

The present study deals with the investigation of the oscillatory morphology of guided slow body MHD modes in inhomogeneous magnetic waveguides that appear in the solar photospheric plasmas in the forms of pores or sunspots. The eigenvalues and eigenfunctions related to these waves in an isothermal plasma are obtained numerically by solving a Sturm-Liouville problem with Dirichlet boundary conditions set at the boundary of the waveguide. Our results show that the inhomogeneities in density (pressure) and magnetic field have a strong influence on the morphology of waves, and higher-order more are sensitive to the presence of inhomogeneity. Our results suggest that he identification of modes just by a simple visual inspection can lead to a misinterpretation of the nature of modes. [ABSTRACT FROM AUTHOR]

We consider the p-Laplace boundary value problem with the Dirichlet boundary condition. A new lower estimate for positive solutions of the problem is obtained. As an application of this new lower estimate, some sufficient conditions for the existence and nonexistence of positive solutions for the p-Laplace problem are obtained. [ABSTRACT FROM AUTHOR]

We present new second-kind integral-equation formulations of the interior and exterior Dirichlet problems for Laplace's equation. The operators in these formulations are both continuous and coercive on general Lipschitz domains in R d , d ≥ 2 , in the space L 2 (Γ) , where Γ denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (1) the Galerkin method converges when applied to these formulations; and (2) the Galerkin matrices are well-conditioned as the discretisation is refined, without the need for operator preconditioning (and we prove a corresponding result about the convergence of GMRES). The main significance of these results is that it was recently proved (see Chandler-Wilde and Spence in Numer Math 150(2):299–371, 2022) that there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which the operators in the standard second-kind integral-equation formulations for Laplace's equation (involving the double-layer potential and its adjoint) cannot be written as the sum of a coercive operator and a compact operator in the space L 2 (Γ) . Therefore there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which Galerkin methods in L 2 (Γ) do not converge when applied to the standard second-kind formulations, but do converge for the new formulations. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence, uniqueness and regularity of weak solutions to the Dirichlet problem for an elliptic equation with a drift $ b $ satisfying $ {\operatorname{div}} b\le 0 $ in $ {\Omega} $. We assume $ b $ belongs to some weak Morrey class which includes in the 3D case, in particular, drifts having a singularity along the axis $ x_3 $ with the asymptotic $ b(x)\sim c/r $, where $ r = \sqrt{x_1^2+x_2^2} $. [ABSTRACT FROM AUTHOR]