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A localized orthogonal decomposition method for semi-linear elliptic problems

Patrick Henning, Axel Målqvist, Daniel Peterseim (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H | log (H) | where H is the coarse mesh size. Without any assumptions...

A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD

Eileen Kammann, Fredi Tröltzsch, Stefan Volkwein (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian....

A variational analysis of a gauged nonlinear Schrödinger equation

Alessio Pomponio, David Ruiz (2015)

Journal of the European Mathematical Society

This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: - Δ u ( x ) + ω + h 2 ( | x | ) | x | 2 + | x | + h ( s ) s u 2 ( s ) d s u ( x ) = | u ( x ) | p - 1 u ( x ) , where h ( r ) = 1 2 0 r s u 2 ( s ) d s . This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for p ( 1 , 3 ) , the functional may be bounded from below or not, depending on ω . Quite surprisingly, the threshold value for ω is explicit. From...

Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents

Manuel del Pino, Fethi Mahmudi, Monica Musso (2014)

Journal of the European Mathematical Society

Let Ω be a bounded domain in n with smooth boundary Ω . We consider the equation d 2 Δ u - u + u n - k + 2 n - k - 2 = 0 in Ω , under zero Neumann boundary conditions, where Ω is open, smooth and bounded and d is a small positive parameter. We assume that there is a k -dimensional closed, embedded minimal submanifold K of Ω , which is non-degenerate, and certain weighted average of sectional curvatures of Ω is positive along K . Then we prove the existence of a sequence d = d j 0 and a positive solution u d such that d 2 | u d | 2 S δ K as d 0 in the sense of measures, where δ K ...

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

Let Ω be a bounded domain of class C 2 in N and let K be a compact subset of Ω . Assume that q ( N + 1 ) / ( N 1 ) and denote by U K the maximal solution of Δ u + u q = 0 in Ω which vanishes on Ω K . We obtain sharp upper and lower estimates for U K in terms of the Bessel capacity C 2 / q , q ' and prove that U K is σ -moderate. In addition we describe the precise asymptotic behavior of U K at points σ K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .

Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions

Futoshi Takahashi (2014)

Mathematica Bohemica

We study the semilinear problem with the boundary reaction - Δ u + u = 0 in Ω , u ν = λ f ( u ) on Ω , where Ω N , N 2 , is a smooth bounded domain, f : [ 0 , ) ( 0 , ) is a smooth, strictly positive, convex, increasing function which is superlinear at , and λ > 0 is a parameter. It is known that there exists an extremal parameter λ * > 0 such that a classical minimal solution exists for λ < λ * , and there is no solution for λ > λ * . Moreover, there is a unique weak solution u * corresponding to the parameter λ = λ * . In this paper, we continue to study the spectral properties of u * and show...

Estimate of the Hausdorff measure of the singular set of a solution for a semi-linear elliptic equation associated with superconductivity

Junichi Aramaki (2010)

Archivum Mathematicum

We study the boundedness of the Hausdorff measure of the singular set of any solution for a semi-linear elliptic equation in general dimensional Euclidean space n . In our previous paper, we have clarified the structures of the nodal set and singular set of a solution for the semi-linear elliptic equation. In particular, we showed that the singular set is ( n - 2 ) -rectifiable. In this paper, we shall show that under some additive smoothness assumptions, the ( n - 2 ) -dimensional Hausdorff measure of singular set...

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), x N , u H ¹ ( N ) , where λ is a positive parameter, a and b are positive functions, while f : is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.

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