Sign changing solutions of semilinear elliptic problems in exterior domains

We prove the existence of a sign changing solution to the semilinear elliptic problem , in an exterior domain Ω having finite symmetries.     

Finite element approximation of elliptic control problems with constraints on the gradient

We consider an elliptic optimal control problem with control constraints and pointwise bounds on the gradient of the state. We present a tailored finite element approximation to this optimal control problem, where the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of the lowest order Raviart–Thomas mixed finite element. Pointwise bounds on the gradient variable are enforced in the elements of the triangulation. Controls are not discretized. Error bounds for control and state are obtained in two and three space dimensions. A numerical example confirms our analytical findings.     

Multiplicity of solutions for a noncooperative elliptic system with critical Sobolev exponent

In this paper, we study a system of elliptic equations by applying the Limit Index Theory. Under some assumptions on nonlinear part, we can obtain the existence of multiple solutions for the equations. The research is supported by NNSF of China (10471024) and Fujian Provincial Natural Science Foundation of China (A0410015).     

Four nontrivial solutions for subcritical exponential equations

We show that a semilinear Dirichlet problem in bounded domains of in presence of subcritical exponential nonlinearities has four nontrivial solutions near resonance. Research supported by the Italian National Project Metodi Variazionali ed Equazioni Differenziali Non Lineari.     

Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents

We are concerned with the semilinear polyharmonic model problem (–)K v = v +v|v| ^s–1 inB,D v|B = 0 for ¦|<-K – 1. HereK ,B is the unit ball in ⁿ,n >2K, is the critical Sobolev exponent. Let ₁ denote the first Dirichlet eigenvalue of (-)^K inB. The existence of a positive radial solutionv is shown for

Stabilized boundary element methods for exterior Helmholtz problems

In this paper we describe and analyze some modified boundary element methods to solve the exterior Dirichlet boundary value problem for the Helmholtz equation. As in classical combined field integral equations also the proposed approach avoids spurious modes. Moreover, the stability of related modified boundary element methods can be shown even in the case of Lipschitz boundaries. The proposed regularization is done based on boundary integral operators which are already included in standard boundary element formulations. Numerical examples are given to compare the proposed approach with other already existing regularized formulations.     

Existence and symmetry of minimizers for nonconvex radially symmetric variational problems

We study functionals of the form

Asymptotic in Sobolev spaces for symmetric Paneitz-type equations on Riemannian manifolds

We describe the asymptotic behaviour in Sobolev spaces of sequences of solutions of Paneitz-type equations [Eq. (E _α ) below] on a compact Riemannian manifold (M, g) which are invariant by a subgroup of the group of isometries of (M, g). We also prove pointwise estimates.     

Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form

On the existence of solutions to a special variational problem

In this paper we establish an existence and regularity result for solutions to the problem

Geometry of quasiminimal phase transitions

We consider the quasiminima of the energy functional

Bifurcation problems for superlinear elliptic indefinite equations with exponential growth

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem

Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem

On approximate solutions to the first initial boundary value problem for the m-Hessian evolution equations

We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness of such solutions for the first initial boundary value problem. Dedicated to Professor Felix Browder     

On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems

We consider the computation of stable approximations to the exact solution of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods

Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M _c such that if solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for . While characterising the possible infinite time blowing-up profile for M  =  M _c , we observe that the long time asymptotics are much more complicated than in the classical Patlak–Keller–Segel system in dimension two. This paper is under the Creative Commons licence Attribution-NonCommercial-ShareAlike 2.5.     

A second order minimality condition for the Mumford-Shah functional

A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on H ¹ ₀(Γ), Γ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of Γ. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.     

A weighted eigenvalue problem for the p-Laplacian plus a potential

Let Δ _p denote the p-Laplacian operator and Ω be a bounded domain in . We consider the eigenvalue problem

Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces

We prove that Lipschitz mappings are dense in the Newtonian–Sobolev classes N ^1,p (X, Y) of mappings from spaces X supporting p-Poincaré inequalities into a finite Lipschitz polyhedron Y if and only if Y is [p]-connected, π ₁(Y) = π ₂(Y) = · · · = π _[p](Y) = 0, where [p] is the largest integer less than or equal to p. This work was supported by the NSF grant DMS-0500966.     

Γ-convergence for incompressible elastic plates

We derive a two-dimensional model for elastic plates as a Γ-limit of three-dimensional nonlinear elasticity with the constraint of incompressibility. The resulting model describes plate bending, and is determined from the isochoric elastic moduli of the three-dimensional problem. Without the constraint of incompressibility, a plate theory was first derived by Friesecke et al. (Comm Pure Appl Math 55:1461–1506, 2002). We extend their result to the case of p growth at infinity with p ϵ [1, 2), and to the case of incompressible materials. The main difficulty is the construction of a recovery sequence which satisfies the nonlinear constraint pointwise. One main ingredient is the density of smooth isometries in W ^2,2 isometries, which was obtained by Pakzad (J Differ Geom 66:47–69, 2004) for convex domains and by Hornung (Comptes Rendus Mathematique 346:189–192, 2008) for piecewise C ¹ domains.