Dirichlet problem
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Exact imposition of boundary conditions with distance functions in\n physics-informed deep neural networks Open
In this paper, we introduce a new approach based on distance fields to\nexactly impose boundary conditions in physics-informed deep neural networks.\nThe challenges in satisfying Dirichlet boundary conditions in meshfree and\nparticle meth…
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A fractional Laplace equation: Regularity of solutions and finite element approximations Open
This paper deals with the integral version of the Dirichlet homogeneous fractional Laplace equation. For this problem weighted and fractional Sobolev a priori estimates are provided in terms of the Holder regularity of the data. By relying…
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Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum Open
For a smooth, bounded domain [math] , [math] , [math] we consider the nonlocal equation ¶\n\n ϵ2s(−Δ)su + u = up in Ω \n\n¶ with zero Dirichlet datum and a small parameter [math] . We construct a family of solutions that concentrate as [ma…
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Multiple solutions for parametric double phase Dirichlet problems Open
We consider a parametric double phase Dirichlet problem. Using variational tools together with suitable truncation and comparison techniques, we show that for all parametric values [Formula: see text] the problem has at least three nontriv…
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Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient Open
In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution s…
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Anisotropic singular double phase Dirichlet problems Open
We consider an anisotropic double phase problem with a reaction in which we have the competing effects of a parametric singular term and a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of po…
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Normalized concentrating solutions to nonlinear elliptic problems Open
We prove the existence of solutions (λ,v)∈R×H1(Ω) of the elliptic problem {−Δv+(V(x)+λ)v=vp in Ω,v>0,∫Ωv2dx=ρ. Any v solving such problem (for some λ) is called a normalized solution, where the normalization is settled in L2(Ω). Here Ω …
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On Dirichlet problem for fractional <i>p</i> -Laplacian with singular non-linearity Open
In this article, we study the following fractional p -Laplacian equation with critical growth and singular non-linearity: ( - Δ p ) s u = λ u - q + u α , u > 0 in Ω , u = 0 in ℝ n ∖ Ω …
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Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball Open
We study the existence and multiplicity of positive radial\nsolutions of the Dirichlet problem for the Minkowski-curvature equation\n$$\n\\begin{cases}\n\\displaystyle\n-{\\rm div}\\bigg( \\frac{\\nabla v} {\\sqrt{1 - |\\nabla v|^2}}\\bigg…
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On a fractional degenerate Kirchhoff-type problem Open
In this paper, we study a highly nonlocal parametric problem involving a fractional-type operator combined with a Kirchhoff-type coefficient. The latter is allowed to vanish at the origin (degenerate case). Our approach is of variational n…
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Existence for fractional Dirichlet boundary value problem under barrier strip conditions Open
In this paper, a fixed-point theorem is used to establish existence results for fractional Dirichlet boundary value problemwhere 1 < α 2, D α x(t) is the conformable fractional derivative, and f :The main condition is sign condition.The me…
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Error bounds for a Dirichlet boundary control problem based on energy spaces Open
In this article, an alternative energy-space based approach is proposed for the Dirichlet boundary control problem and then a finite-element based numerical method is designed and analyzed for its numerical approximation. A priori error es…
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Overcoming Order Reduction in Diffusion-Reaction Splitting. Part 1: Dirichlet Boundary Conditions Open
For diffusion-reaction equations employing a splitting procedure is\nattractive as it reduces the computational demand and facilitates a parallel\nimplementation. Moreover, it opens up the possibility to construct second-order\nintegrators…
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Existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations Open
In this paper, we establish the existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations in a ball. Under some suitable local growth conditions for nonlinearity, several new results…
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Quintic NLS in the exterior of a strictly convex obstacle Open
We consider the defocusing energy-critical nonlinear Schr\"odinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions. For the initial-value problem with Dirichlet boundary condition we prove global w…
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On the existence of multiple solutions for a partial discrete Dirichlet boundary value problem with mean curvature operator Open
Apartial discrete Dirichlet boundary value problem involving mean curvature operator is concerned in this paper. Under proper assumptions on the nonlinear term, we obtain some feasible conditions on the existence of multiple solutions by t…
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The Cauchy–Dirichlet problem for a general class of parabolic equations Open
We prove regularity results such as interior Lipschitz regularity and boundary continuity for the Cauchy–Dirichlet problem associated to a class of parabolic equations inspired by the evolutionary p -Laplacian, but extending it at a wide s…
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Dirichlet problems for fully anisotropic elliptic equations Open
The existence of a non-trivial bounded solution to the Dirichlet problem is established for a class of nonlinear elliptic equations involving a fully anisotropic partial differential operator. The relevant operator depends on the gradient …
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A Fourth Order Singular Elliptic Problem Involving $p$-biharmonic Operator Open
In this paper, a fourth order singular elliptic problem involving $p$-biharmonic operator with Dirichlet boundary condition is considered. The existence of at least one weak solution is proved in two different cases of the nonlinear term a…
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A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian Open
We consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the ( p , q )-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The ma…
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Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution Open
We propose a two-scale finite element method for the Monge-Ampère equation with Dirichlet boundary condition in dimension $d\ge 2$ and prove that it converges to the viscosity solution uniformly. The method is inspired by a finite differen…
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Resonant double phase equations Open
We consider a double phase Dirichlet equation with a reaction which is asymptotically as x→±∞, resonant with respect to the first eigenvalue of a related eigenvalue problem. Using variational tools together with Morse theoretic arguments, …
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A-priori bounds and existence for solutions of weighted elliptic equations with a convection term Open
We investigate weighted elliptic equations containing a convection term with variable exponents that are subject to Dirichlet or Neumann boundary condition. By employing the De Giorgi iteration and a localization method, we give a-priori b…
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Enforcing Dirichlet boundary conditions in physics-informed neural networks and variational physics-informed neural networks Open
In this paper, we present and compare four methods to enforce Dirichlet boundary conditions in Physics-Informed Neural Networks (PINNs) and Variational Physics-Informed Neural Networks (VPINNs). Such conditions are usually imposed by addin…
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Boundary value problem with fractional <i>p</i> -Laplacian operator Open
The aim of this paper is to obtain the existence of solution for the fractional p -Laplacian Dirichlet problem with mixed derivatives t D T α (| 0 D t α u ( t )| p -2 0 D t α u ( t )) = f ( t , u ( t )), t ∈ [0, T ], u (0) = u ( T ) = 0, w…
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A new HDG method for Dirichlet boundary control of convection diffusion PDEs II: Low regularity Open
The work of the first author was supported by the National Natural Science Foundation of China under grants 11671391 and 91530204. The work of the second author was partially supported by the DIG and FY 2018 ASR+1 Program at Oklahoma State…
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Noncoercive resonant (<i>p</i>,2)-equations with concave terms Open
We consider a nonlinear Dirichlet problem driven by the sum of a p -Laplace and a Laplacian (a (p,2) {(p,2)} -equation). The reaction exhibits the competing effects of a parametric concave term plus a Caratheodory perturbation which is res…
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The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains Open
We study the interior regularity of solutions to the Dirichlet problem Lu=g in Ω, u=0 in Rn∖Ω, for anisotropic operators of fractional typeLu(x)=∫0+∞dρ∫Sn−1da(ω)2u(x)−u(x+ρω)−u(x−ρω)ρ1+2s. Here, a is any measure on Sn−1 (a prototype exampl…
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A Lazer-McKenna type problem with measures Open
In this paper we are concerned with a general singular Dirichlet boundary value problem whose model is the following $$ \begin{cases} -Δu = \fracμ{u^γ} & \text{in}\ Ω, u=0 &\text{on}\ \partialΩ, u>0 &\text{on}\ Ω\,. \end{cases} $$ Here $μ$…
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A Liouville type theorem for Lane–Emden systems involving the fractional Laplacian Open
We establish a Liouville type theorem for the fractional Lane-Emden system: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=v^q&{\rm in}\,\,\R^N,\\ (-\Delta)^\alpha v=u^p&{\rm in}\,\,\R^N, \end{array} \right. \end{eq…