Semilinear Elliptic Equations Research Articles

The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs

This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under [Formula: see text]-Gevrey assumptions on the residual equation, we establish [Formula: see text]-Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.

On periodic and compactly supported least energy solutions to semilinear elliptic equations with non-Lipschitz nonlinearity

We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: − Δ u = λ u p − u q in R N + 1 , where 0 < q < p < 1 and λ ∈ R. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter λ, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p, q. Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space R N + 1 .

Adaptive Multi-level Algorithm for a Class of Nonlinear Problems

Abstract In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of our theory, we consider the pseudostress-velocity formulation of Navier–Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.

Cauchy problem for a semilinear elliptic equation with contaminated coefficients

In this paper, we assume that Ω is a bounded domain of RN with smooth boundary. Let A:D(A)→L2(Ω) be a self-adjoint operator defined on a dense subspace D(A)⊂L2(Ω) with an orthonormal basis of eigenfunctions in L2(Ω). For a constant Y>0, giving the function F:Ω×[0,Y]×R→R, we consider the problem of finding a function u:Ω×[0,Y]→R such that −τ(y)Au(x,y)+uyy(x,y)=F(x,y,u(x,y)),x∈Ω,0<y<Y,u(x,0)=f(x),uy(x,0)=g(x),x∈Ω, where f, g∈L2(Ω) are given. In many practical cases, the function τ:[0,Y]→(0,∞) is contaminated with noise and it can only be approximated by an experimentally observable function μ:[0,Y]→(0,∞) such that sup0⩽y⩽Y∑m=02dmτdym(y)−dmμdym(y)⩽δfor a δ>0. Similarly, for ɛ>0, the initial data f,g are often perturbed by fɛ,gɛ as follows. ‖f−fɛ‖L2(Ω)+‖g−gɛ‖L2(Ω)⩽ɛ.The Cauchy problem with these contaminated data is nonlinear and ill-posed. Due to the usefulness in physics and other fields, we consider a regularization for the problem.

A free boundary problem for discontinuous semilinear elliptic equations in convex ring

In this paper, we consider the following free boundary problem: ( P ) { Δ u = λ ϕ ( x ) ∑ i = 1 n H ( u − μ i ) in Ω = Ω 2 ∖ Ω ¯ 1 , u = 0 on ∂ Ω 2 , u = M on ∂ Ω 1 . The domain Ω is a convex ring where Ω 1 and Ω 2 are bounded convex domains in R N , N ≥ 2 , H is the Heaviside step function, λ , M , μ i ( i = 1 , . . , n ) are given positive real parameters and ϕ is a given function. We show that under suitable conditions, there exists a solution to problem (P) with a convex level set. We derive also an interesting result for the convexity of the set which is delimited by the free boundary in obstacle problem when the nonlinearity is discontinuous. At last, a detailed analysis is given by a construction of function ϕ which shows that there are more solutions.

Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon

In an open, bounded Lipschitz polygon Ω⊂R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset \\mathbb {R}^2$$\\end{document}, we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document}. The boundary conditions on each edge of ∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega $$\\end{document} are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in H1(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1(\\Omega )$$\\end{document}, quantized tensor formats and certain deep neural networks.

On oscillating radial solutions for non-autonomous semilinear elliptic equations

&lt;abstract&gt;&lt;p&gt;We consider semilinear elliptic equations of the form $ \Delta u + f(|x|, u) = 0 $ on $ {\mathbb{R}}^{N} $ with $ f(|x|, u) = q(|x|)g(u) $. These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity $ f $, there exists an oscillating radial solution converging to a zero of the function $ g $. We also study the oscillating and limiting behavior of this solution.&lt;/p&gt;&lt;/abstract&gt;

On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs

We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation (PDE) which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc(ℝ). This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. The convergence of the resulting algorithm is proven in the function space setting. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation functions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero. The paper concludes with numerical experiments which confirm the theoretical findings.

Properties of the least action level and the existence of ground state solution to fractional elliptic equation with harmonic potential

In this article we consider the following fractional semilinear elliptic equation \[(-\Delta)^su+|x|^2u =\omega u+|u|^{2\sigma}u \quad \text{ in } \mathbb{R}^N,\] where \(s\in (0,1)\), \(N\gt 2s\), \(\sigma\in (0,\frac{2s}{N-2s})\) and \(\omega\in (0, \lambda_1)\). By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result.

A Reduced Order Schwarz Method for Nonlinear Multiscale Elliptic Equations Based on Two-Layer Neural Networks

Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts, including solution of partial differential equations (PDEs). We describe a solver for multiscale fully nonlinear elliptic equations that makes use of domain decomposition, an accelerated Schwarz framework, and two-layer neural networks to approximate the boundary-to-boundary map for the subdomains, which is the key step in the Schwarz procedure. Conventionally, the boundary-to-boundary map requires solution of boundary-value elliptic problems on each subdomain. By leveraging the compressibility of multiscale problems, our approach trains the neural network offline to serve as a surrogate for the usual implementation of the boundary-to-boundary map. Our method is applied to a multiscale semilinear elliptic equation and a multiscale $p$-Laplace equation. In both cases we demonstrate significant improvement in efficiency as well as good accuracy and generalization performance.

Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities

Let $\Omega \subset \mathbb {R}^N$ ( $N\geq 3$ ) be a $C^2$ bounded domain and $\Sigma \subset \partial \Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$ , $0\leq k \leq N-1$ . We assume that $\Sigma = \{0\}$ if $k = 0$ and $\Sigma =\partial \Omega$ if $k=N-1$ . Let $d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$ and $L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$ , where $\mu \in {\mathbb {R}}$ . We study boundary value problems ( $P_\pm$ ) $-{L_\mu} u \pm |u|^{p-1}u = 0$ in $\Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)=\nu$ on $\partial \Omega$ , where $p&gt;1$ , $\nu$ is a given measure on $\partial \Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)$ denotes the boundary trace of $u$ associated to $L_\mu$ . Different critical exponents for the existence of a solution to ( $P_\pm$ ) appear according to concentration of $\nu$ . The solvability for problem ( $P_+$ ) was proved in [3, 29] in subcritical ranges for $p$ , namely for $p$ smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of $-L_\mu$ , we provide conditions on $\nu$ expressed in terms of capacities for the existence of a (unique) solution to ( $P_+$ ) in supercritical ranges for $p$ , i.e. for $p$ equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to ( $P_-$ ) under a smallness assumption on $\nu$ .

Bubbling phenomenon for semilinear Neumann elliptic equations of critical exponential growth

Bubbling phenomenon for semilinear Neumann elliptic equations of critical exponential growth

On semilinear elliptic equation with negative exponent arising from a closed MEMS model

On semilinear elliptic equation with negative exponent arising from a closed MEMS model

Semilinear optimal control with Dirac measures

Abstract The purpose of this work is to study an optimal control problem for a semilinear elliptic partial differential equation with a linear combination of Dirac measures as a forcing term; the control variable corresponds to the amplitude of such singular sources. We analyze the existence of optimal solutions and derive first- and, necessary and sufficient, second-order optimality conditions. We develop a solution technique that discretizes the state and adjoint equations with continuous piecewise linear finite elements; the control variable is already discrete. We analyze the convergence properties of discretizations and obtain, in two dimensions, an a priori error estimate for the underlying approximation of an optimal control variable.

Convergence of adaptive two-grid weak Galerkin finite element methods for semilinear elliptic differential equations

In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (Pj(T),Pℓ(∂T),RTj(T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j=l=0 are provided to support the theoretical results.

A class of elliptic problems driven by a mixed local-nonlocal operator

In this paper, we consider a class of elliptic problems driven by a mixed local-nonlocal operator. By estimating the critical groups and using Morse theory, the existence of nontrivial solutions is obtained in different cases. Our results extend some classical theorems for semilinear elliptic equations to the mixed local-nonlocal setting.

Existence of the solution for a class of the semilinear degenerate elliptic equation involving the Grushin operator in R2$\mathbb {R}^2$: The interaction between Grushin's critical exponent and exponential growth

AbstractIn this work, we study the existence of the solution for a class of semilinear degenerate elliptic equations in the whole space involving the Grushin operator and a nonlinearity that involves the Grushin's critical growth and the exponential growth. The existence of at least one nontrivial weak solution is done by combining the mountain‐pass theorem, Trudinger–Moser inequality, and a version of a result due to Lions for the exponential growth in in the corresponding weighted Sobolev space.

Superharmonic functions related to convexity estimates for a class of semilinear elliptic problems

In this paper, we consider homogeneous Dirichlet problems for a class of semilinear elliptic equations involving the Laplacian with power-type nonlinearities in bounded convex domains of Rn, and find some superharmonic functions which can be used to give convexity estimates for the solutions directly.

Necessary and sufficient conditions for the existence of large solutions to semilinear elliptic equations with gradient terms

Our purpose in this paper is to carry out a comprehensive investigation into the question of existence and non-existence of large solutions to semilinear elliptic equations with non-linear gradient terms in bounded domains of the Euclidean space. Based on new Keller-Osserman type integrability criteria, we provide necessary and sufficient conditions for the existence of large solutions. The novelty in our study rests on the fact that the Keller-Osserman type conditions we employ take both the growth rates of the gradient term as well as those of the first order non-linearity terms into account. When the gradient term does not appear, our criteria reduces to the well-known Keller-Osserman condition.

Hölder stability for a semilinear elliptic inverse problem

We are concerned with the problem of determining the nonlinear term in a semilinear elliptic equation by boundary measurements. Precisely, we improve [5, Theorem 1.3], where a logarithmic type stability estimate was proved. We show actually that we have a Hölder stability estimate with less boundary measurements and less regular nonlinearities. We establish our stability inequality by following the same method as in [4]. This method consists in constructing special solutions vanishing on a subboundary of the domain.