Three-term asymptotics for the boundary layers of semilinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in R^N (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution u_λ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of u_λ with the exact second term and the ‘optimal’ estimate of the third term.     

A bifurcation problem governed by the boundary condition I

We deal with positive solutions of Δu = a(x)u ^p in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ₁ where, roughly speaking, σ₁ is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile of the solution as λ → σ₁. Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D. Rossi). J.D. Rossi is a member of CONICET.     

Marching schemes for inverse acoustic scattering problems

Summary For the numerical solution of inverse Helmholtz problems the boundary value problem for a Helmholtz equation with spatially variable wave number has to be solved repeatedly. For large wave numbers this is a challenge. In the paper we reformulate the inverse problem as an initial value problem, and describe a marching scheme for the numerical computation that needs only n² log n operations on an n × n grid. We derive stability and error estimates for the marching scheme. We show that the marching solution is close to the low-pass filtered true solution. We present numerical examples that demonstrate the efficacy of the marching scheme.     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which depends smoothly (in W ^2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.     

The heterogeneous Allen-Cahn equation in a ball: Solutions with layers and spikes

Let u? be a single layered radially symmetric unstable solution of the Allen-Cahn equation −?²Δu=u(u−a(|x|))(1−u) over the unit ball with Neumann boundary conditions. Based on our estimate of the small eigenvalues of the linearized eigenvalue problem at u? when ? is small, we construct solutions of the form u?+v?, with v? non-radially symmetric and close to zero in the unit ball except near one point x₀ such that |x₀| is close to a nondegenerate critical point of a(r). Such a solution has a sharp layer as well as a spike.     

Existence and regularity of the solution of a mixed boundary value problem for the Keldysh equation with a nonlinear absorption term

The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y>0 and is degenerate at the line y=0 in R². Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful.     

Linear and nonlinear abstract equations with parameters

This paper focuses on nonlocal boundary value problems for linear and nonlinear abstract elliptic equations in Banach spaces. Here equations and boundary conditions contain certain parameters. The uniform separability of the linear problem and the existence and uniqueness of maximal regular solution of nonlinear problem are obtained in L_p spaces. For linear case the discreteness of spectrum of corresponding parameter dependent differential operator is obtained. The behavior of solution when the parameter approaches zero and its smoothness with respect to the parameter is established. Moreover, we show the estimate for analytic semigroups in terms of interpolation spaces. This fact can be used to obtain maximal regularity properties for abstract boundary value problems.     

Existence and non-existence of solutions for a class of Monge-Ampère equations

We study the boundary value problems for Monge-Ampère equations: detD²u=e−u in Ω⊂Rn, n?1, u_|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD²u=e−tu in Ω, u_|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure.     

Multiple Positive Solutions for Eigenvalue Problems of Hemivariational Inequalities

We study a nonlinear eigenvalue problem with a nonsmooth potential. The subgradients of the potential are only positive near the origin (from above) and near +∞. Also the subdifferential is not necessarily monotone (i.e. the potential is not convex). Using variational techniques and the method of upper and lower solutions, we establish the existence of at least two strictly positive smooth solutions for all the parameters in an interval. Our approach uses the nonsmooth critical point theory for locally Lipschitz functions. A byproduct of our analysis is a generalization of a result of Brezis-Nirenberg (CRAS, 317 (1993)) on H¹₀ versus C¹₀ minimizers of a C¹-functional.     

Integral representation for the solution of the stationary Schrödinger equation in a cone

In this paper, we consider the Dirichlet problem for the stationary Schrödinger equation in a cone with continuous boundary data. For a solution u of the stationary Schrödinger equation in a cone, we prove that if its positive part u⁺ satisfying a slowly growing condition, then its negative part u^? can also be dominated by a similar slowly growing condition. Meanwhile, u can be represented by its integral in the boundary of the cone.     

Large solutions to the p-Laplacian for large p

In this work we consider the behaviour for large values of p of the unique positive weak solution u _p to Δ _p u = u ^q in Ω, u = +∞ on , where q > p − 1. We take q = q(p) and analyze the limit of u _p as p → ∞. We find that when q(p)/p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of with u = +∞ on . If Q = 1 then solutions go to ∞ in the whole Ω and when Q = ∞ solutions converge to 1 uniformly in compact subsets of Ω, hence the boundary blow-up is lost in the limit.     

On the number of nodal solutions to a singularly perturbed Neumann problem

We show that for ε small, there are arbitrarily many nodal solutions for the following nonlinear elliptic Neumann problem where Ω is a bounded and smooth domain in ℝ² and f grows superlinearly. (A typical f(u) is f(u)= a₁ u₊^p – a₁ u_-^p, a₁, a₂ >0, p, q>1.) More precisely, for any positive integer K, there exists ε_K>0 such that for 0<ε<ε_K, the above problem has a nodal solution with K positive local maximum points and K negative local minimum points. This solution has at least K+1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on ∂Ω. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed.     

Exterior nonlinear Neumann problem

We consider the solvability of the Neumann problem for equation (1.1) in exterior domains in both cases: subcritical and critical. We establish the existence of least energy solutions. In the subcritical case the coefficient b(x) is allowed to have a potential well whose steepness is controlled by a parameter λ > 0. We show that least energy solutions exhibit a tendency to concentrate to a solution of a nonlinear problem with mixed boundary value conditions.     

Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent

We consider the boundary value problem Δu+up=0 in a bounded, smooth domain Ω in R² with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p→∞. In particular, for a nonsimply connected domain such a solution exists for any given m?1.     

Structure of boundary blow-up solutions for quasi-linear elliptic problems II: small and intermediate solutions

The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δ_pu=λf(u), u=∞ on ∂Ω, 1<p<∞, is studied in a bounded smooth domain , for a class of nonlinearities f∈C¹((0,∞)?{z₂})∩C⁰[0,∞) satisfying f(0)=f(z₁)=f(z₂)=0 with 0<z₁<z₂, f<0 in (0,z₁)∪(z₂,∞), f>0 in (z₁,z₂). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2.     

Application of the G class of functions in the parabolic class

In this paper,the application of the G class of functions in the parabolic class is considered. The regularity of the solution for the first boundary value problem of parabolic equation in divergence form is proved.     

Schauder estimates for a class of second order elliptic operators on a cube

We consider a class of second order elliptic operators on a d-dimensional cube S_d. We prove that if the coefficients are of class C^k+δ(S_d), with k=0,1 and δ∈(0,1), then the corresponding elliptic problem admits a unique solution u belonging to C^k+2+δ(S_d) and satisfying non-standard boundary conditions involving only second order derivatives.     

The harmonic dirichlet problem for a cracked domain with jump conditions on cracks

The harmonic problem in a cracked domain is studied in R ^m , m?>?2. The boundary of the domain is assumed to be nonsmooth, while cracks are smooth. The Dirichlet condition is specified on the boundary of the domain. Jumps of the unknown function and its normal derivative are specified on the cracks. Uniqueness and solvability results are obtained. The problem is reduced to the uniquely solvable integral equation, its solution is given explicitely in the form of a series. The estimates of the solution of the problem depending on the boundary data are obtained.     

Multiple solutions of semilinear degenerate elliptic boundary value problems

The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet problem and the Robin problem. The approach here is based on the super‐sub‐solution method in the degenerate case, and is distinguished by the extensive use of an L^p Schauder theory elaborated for second‐order, elliptic differential operators with discontinuous zero‐th order term. By using Schauder's fixed point theorem, we prove that the existence of an ordered pair of sub‐ and supersolutions of our problem implies the existence of a solution of the problem. The results extend an earlier theorem due to Kazdan and Warner to the degenerate case. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Isolated singularities of Monge-Ampère equations

Letz=z(x, y) be a real-valued twice continuously differentiable solution of the elliptic Monge-Ampère equationAr+2Bs+Ct+rt – s ²=E in the punctured disk 0<(x–x ₀)²+(y–y ₀)²<². Assume thatq is continuous at (x₀, y₀). Our aim is to give sufficient conditions on the coefficientsA,..., E which ensure that the singularity (x ₀,y ₀) is removable. This generalizes an earlier result of Jörgens (Math. Ann. 129 (1955), 330–344).