Optimal control for a parabolic problem in a domain with highly oscillating boundary

In this article we study the homogenization of an optimal control problem for a parabolic equation in a domain with highly oscillating boundary. We identify the limit problem, which is an optimal control problem for the homogenized equation and with a different cost functional.     

On a variational approach to some non-local boundary value problems

We treat by variational methods some nonlinear elliptic b.v.p. that involve a non-local term. Our main tools are the Mountain Pass Theorem of Ambrosetti and Rabinowitz and minimization procedures.     

A Faber-Krahn inequality for the Laplacian with Generalised Wentzell boundary conditions

We give an extension of the Faber-Krahn inequality to the Laplacian Δ on bounded Lipschitz domains , with generalised Wentzell boundary conditions on ∂Ω, where β, γ are nonzero real constants. We prove that when β, γ > 0, the ball B minimises the first eigenvalue with respect to all Lipschitz domains Ω of the same volume as B, and that B is the unique minimiser amongst C ²-domains. We also consider β, γ not both positive, and slightly extend what is known about the associated Wentzell operator and its resolvent in addition to considering an analogue of the Faber-Krahn inequality. This is based on the recent extension of the Faber-Krahn inequality to the Robin Laplacian. We also give a version of Cheeger’s inequality for the Wentzell Laplacian when β, γ > 0.      

A Lower Bound for the Height of a Rational Function at <Emphasis Type="Italic">S</Emphasis>-unit Points

Let a,b be given, multiplicatively independent positive integers and let >0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(n) for g.c.d.(aⁿ–1, bⁿ–1); shortly afterwards we generalized this to the estimate g.c.d.(u–1,v–1)v) for multiplicatively independent S-units u,vZ. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g.c.d. to pairs of rational functions other than {u–1,v–1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G_m².     

On low eigenvalues of the Laplacian with mixed boundary conditions

By means of the so-called α-symmetrization we study the eigenvalue problem for the Laplace operator with mixed boundary conditions. We obtain various bounds for combinations of the low eigenvalues and some sharp comparison results for the first eigenfunction in terms of Bessel functions.     

Ruled Real Hypersurfaces in a Nonflat Quaternionic Space Form

In this paper we construct many ruled real hypersurfaces in a nonflat quaternionic space form systematically, and in particular give an example of a homogeneous ruled real hypersurface in a quaternionic hyperbolic space. In the second half of this paper we characterize them by investigating the extrinsic shape of their geodesics. We also characterize curvature-adapted real hypersurfaces in nonflat quaternionic space forms from the same viewpoint.The first author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 14540075), Ministry of Education, Science, Sports and Culture.The second author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 14540080), Ministry of Education, Science, Sports and Culture.     

Multiple Positive Radial Solutions of Elliptic Equations in an Exterior Domain

Several existence theorems on multiple positive radial solutions of the elliptic boundary value problem in an exterior domain are obtained by using the fixed point index theory. Our conclusions are essential improvements of the results in [7], [10] and [13].     

Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with an oscillating boundary

The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating boundary is analyzed. The leading terms of the asymptotic expansions for eigenelements are constructed, and the asymptotics are substantiated for simple eigenvalues. The text was submitted by the authors in English.     

Concentration of low energy extremals: Identification of concentration points

We study the variational problem where , is a bounded domain, , F satisfies $0\leq F|t|\leq \alpha |t|^{2^*}$ and is upper semicontinuous. We show that to second order in the value only depends on two ingredients. The geometry of enters through the Robin function (the regular part of the Green's function) and F enters through a quantity which is computed from (radial) maximizers of the problem in . The asymptotic expansion becomes Using this we deduce that a subsequence of (almost) maximizers of must concentrate at a harmonic center of : i.e., , where is a minimum point of . Received: 24 January 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001     

Oblique boundary value problems for equations of Monge-Ampère type

We prove the existence of classical solutions of elliptic equations of Monge-Ampère type subject to a semilinear oblique boundary condition which is a perturbation of the Neumann boundary condition. Our techniques also allow us to treat fully nonlinear strictly oblique boundary conditions satisfying a concavity condition. Examples show that the above restrictions on the boundary condition are generally necessary for the existence of classical solutions. Received May 22, 1996 / Accepted April 10, 1997     

Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equation

We analyze the well-posedness of the initial value problem for the dissipative quasi-geostrophic equations in the subcritical case. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions. While the only small self-similar solution in the strong space is the null solution, infinitely many self-similar solutions do exist in weak- spaces and in a recently introduced [7] space of tempered distributions. The asymptotic stability of solutions is obtained in both spaces, and as a consequence, a criterion of self-similarity persistence at large times is obtained.     

A degenerate diffusion problem with dynamical boundary conditions Diffusion problem with dynamical boundary conditions

The purpose of this paper is to study the existence, the uniqueness and the limit in , as of solutions of general initial-boundary-value problems of the form and in a bounded domain with dynamical boundary conditions of the form Received: 5 December 2000 / Revised version: 20 November 2001 / Published online: 4 April 2002     

Solutions and multiple solutions for problems with the <Emphasis Type="Italic">p</Emphasis>-Laplacian

In this paper we consider two nonlinear elliptic problems driven by the p-Laplacian and having a nonsmooth potential (hemivariational inequalities). The first is an eigenvalue problem and we prove that if the parameter λ < λ₂ = the second eigenvalue of the p-Laplacian, then there exists a nontrivial smooth solution. The second problem is resonant both near zero and near infinity for the principal eigenvalue of the p-Laplacian. For this problem we prove a multiplicity result. Our approach is variational based on the nonsmooth critical point theory. Second author is Corresponding author.     

Regularity in a Singular Biharmonic Dirichlet Problem

We consider the singular biharmonic equation with Dirichlet boundary conditions u = f₀ and ∂_nu = f₁ on . In our setup the boundary values f_j (j = 0,1) are elements in two homogeneous Banach spaces B_j (j = 0,1) on . We give a sufficient condition on the spaces B_j (j = 0,1) to ensure that the solution u of this Dirichlet problem has the appropriate boundary values f_j (j = 0,1) in the sense of convergence in spaces B_j (j = 0,1). Our results also apply in the unweighted case.     

A Composition Theorem for Multiple Summing Operators

We prove that the composition S(u₁, …, u_n) of a multilinear multiple 2-summing operator S with 2-summing linear operators u_j is nuclear, generalizing a linear result of Grothendieck. Both authors were partially supported by DGICYT grant BMF2001-1284.     

The multiplicity of solutions in non-homogeneous boundary value problems

We use a method recently devised by Bolle to establish the existence of an infinite number of solutions for various non-homogeneous boundary value problems. In particular, we consider second order systems, Hamiltonian systems as well as semi-linear partial differential equations. The non-homogeneity can originate in the equation but also from the boundary conditions. The results are more satisfactory than those obtained by the standard “Perturbation from Symmetry” method that was developed – in various forms – in the early eighties by Bahri–Berestycki, Struwe and Rabinowitz. Received: 13 August 1998 / Revised version: 6 July 1999     

Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks

We study the Cahn-Hilliard equation in a bounded smooth domain without any symmetry assumptions. We prove that for any fixed positive integer K there exist interior K–spike solutions whose peaks have maximal possible distance from the boundary and from one another. This implies that for any bounded and smooth domain there exist interior K–peak solutions. The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy to finite dimensions (Lemma 5.5) with variables which are closely related to the location of the peaks. We do not assume nondegeneracy of the points of maximal distance to the boundary but can do with a global condition instead which in many cases is weaker. Received March 5, 1999 / Accepted June 11, 1999