Nonexistence of single blow-up solutions for a nonlinear Schrödinger equation involving critical Sobolev exponent

In this paper we shall consider the critical elliptic equation where and a(x) is a real continuous, non negative function, not identically zero. By using a local Pohozaev identity, we show that problem (0.1) does not admit a family of solutions which blows-up and concentrates as at some zero point x₀ of a(x) if the order of flatness of the function a(x) at x₀ is      

A free boundary problem for $\infty$–Laplace equation

We consider a free boundary problem for the p-Laplacian describing nonlinear potential flow past a convex profile K with prescribed pressure on the free stream line. The main purpose of this paper is to study the limit as of the classical solutions of the problem above, existing under certain convexity assumptions on a(x). We show, as one can expect, that the limit solves the corresponding potential flow problem for the -Laplacian in a certain weak sense, strong enough however, to guarantee uniqueness. We show also that in the special case the limit is given by the distance function. Received: 10 October 2000 / Accepted: 23 February 2001 / Published online: 19 October 2001     

Infinitely many solutions of dirichlet problem for <Emphasis Type="Italic">p</Emphasis>-mean curvature operator

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R ⁿ (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).     

On a semigroup generated by a convex combination of two Feller generators

Let be a locally compact Hausdorff space. Let A and B be two generators of Feller semigroups in with related Feller processes {X _A (t), t ≥ 0} and {X _B (t), t ≥ 0} and let α and β be two non-negative continuous functions on with α + β = 1. Assume that the closure C of C ₀ = αA + βB with generates a Feller semigroup {T _C (t), t ≥ 0} in . It is natural to think of a related Feller process {X _C (t), t ≥ 0} as that evolving according to the following heuristic rules. Conditional on being at a point , with probability α(p) the process behaves like {X _A (t), t ≥ 0} and with probability β(p) it behaves like {X _B (t), t ≥ 0}. We provide an approximation of {T _C (t), t ≥ 0} via a sequence of semigroups acting in that supports this interpretation. This work is motivated by the recent model of stochastic gene expression due to Lipniacki et al. [17].     

Differentiability preserving properties of a class of semigroups

It is well known that for a large class of Markov process the associated semi-group T(t)f(x)=f(y)P(t,x;dy) satisfies the Kolmogorov backward differential equation, that is, if u(t,x)=T(t)f(x) then and .In this paper we are considering the opposite problem: given the diffusion and drift coefficients we study the differentiability preserving properties of the semigroup T(t) having as infinitesimal generator .More specifically, for a large class of functions a(x) and b(x), we will prove for k=0, ..., 3 the existence of T(t) such that T(t): C ^k (I) C ^k (I) and the existence of a constant _k such that |T(t)f| _k |f| _k exp ( _k t) for fC ^k (I). Moreover an explicit expression of _k in terms of the coefficients a(x) and b(x) is obtained. As a side result we obtain the necessity of the boundary conditions imposed.This paper is a revised version of the author's Ph. D. dissertation at University of Massachusetts under W. Rosenkrantz     

An optimal control problem for microwave heating

In this paper, we study an optimization problem for a microwave/induction heating process. The cost function is defined such that the temperature profile at the final stage has a relative uniform distribution in the field. The control variable is the applied electric field on the boundary. We show that there exists an optimal electric field which minimizes the cost function. Moreover, a necessary condition for a special case is also derived.     

Stationary layered solutions in {\mathbb{R}}^2 for a class of non autonomous Allen-Cahn equations

We consider a class of non autonomous Allen-Cahn equations where is a multiple-well potential and is a periodic, positive, non-constant function. We look for solutions to (0.1) having uniform limits as corresponding to minima of W. We show, via variational methods, that under a nondegeneracy condition on the set of heteroclinic solutions of the associated ordinary differential equation the equation (0.1) has solutions which depends on both the variables x andy. In contrast, when a is constant such nondegeneracy condition is not satisfied and all such solutions are known to depend only on x. Received April 16, 1999 / Accepted October 1, 1999 / Published online June 28, 2000     

Self-similar solutions for the anisotropic affine curve shortening problem

Similarity between the roles of the group on the equation for self-similar solutions of the anisotropic affine curve shortening problem and of the conformal group of on the Nirenberg problem for prescribed scalar curvature is explored. Sufficient conditions for the existence of affine self-similar curves are established. Received June 26, 1999 / Accepted January 28, 2000 / Published online December 8, 2000     

Fully nonlinear elliptic equations and the dini condition

Abstract

We investigate transmission problems with strongly Lipschitz interfaces for the Dirac equation by establishing spectral estimates on an associated boundary singular integral operator, the rotation operator. Using Rellich estimates we obtain angular spectral estimates on both the essential and full spectrum for general bi-oblique transmission problems. Specializing to the normal transmission problem, we investigate transmission problems for Maxwell's equations using a nilpotent exterior/interior derivativeoperator. The fundamental commutation properties for this operator with the two basic reflection operators are proved. We show how the L ₂spectral estimates are inherited for the domain of the exterior/interior derivative operator and prove some complementary eigenvalue estimates. Finally we use a general algebraic theorem to prove a regularity property needed for Maxwell's equations.     

Single-point condensation phenomena for a four-dimensional biharmonic Ren–Wei problem

We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E _p ): as p gets large, where Ω is a smooth bounded domain in R ⁴ . In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc. AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky pattern, under the assumption that the domain is convex.     

Existence and uniqueness results for large solutions of general nonlinear elliptic equations

We study under what condition there exists a solution of -u+f(u)=0 in a domain which blows-up on the boundary, independently of the regularity of the boundary, and we provide criteria for uniqueness. We apply our results to the case f(u)= e^au.     

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.     

Positive solutions of asymptotically linear Schrödinger-Poisson systems with a radial potential vanishing at infinity

In this paper, we study the Schrödinger-Poisson system

(SP)     

Martin–Kuramochi boundary and reflecting symmetric diffusion

In this paper, we will give sufficient conditions for the existence of the reflecting diffusion process on a locally compact space. In constructing reflecting diffusion process, we consider the corresponding Martin–Kuramochi boundary as the reflecting barrier and introduce the notion of strong (ℰ, u)-Caccioppoli set. Our method covers reflecting diffusion processes with diffusion coefficient degenerating on the boundary. Received: 23 June 1997 / Revised version: 28 September 1991/ Published online: 14 June 2000     

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.     

Existence of bounded solutions for nonlinear elliptic equations in unbounded domains

In this paper we study the existence of bounded weak solutions for some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the p-laplacian operator, and the lower order terms, which depend on the solution u and its gradient u, have a power growth of order p–1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity.     

The effect of concentrating potentials in some singularly perturbed problems

The equation with boundary Dirichlet zero data is considered in a bounded domain . Under the assumption that concentrates, as , round a manifold and that f is a superlinear function, satisfying suitable growth assumptions, the existence of multiple distinct positive solutions is proved. Received: 19 December 2000 / Accepted: 8 May 2001 / Published online: 5 September 2002     

Nonlinear elliptic equations with subhomogeneous potentials

We prove the existence of nonnegative symmetric solutions to the semilinear elliptic equation

     

Homogenization in general periodically perforated domains by a spectral approach

In this article we study the asymptotic behaviour as tends to 0 of the Neumann problem $-\Delta u_\epsilon+u_\epsilon=\epsilon$-periodic bounded open set of . The period cell of is equal to where is a regular open subset of the d-dimensional torus. We prove that if there exists a smallest integer such that the n-th non-zero eigenvalue of the spectral problem in satisfies , the limiting problem is a linear system of second order p.d.e.'s, of size n. By this spectral approach we extend in the periodic framework a result due to Khruslov without making strong geometrical assumptions on the perforated domain . Received: 20 December 2000 / Accepted: 11 May 2001 / Published online: 19 October 2001