Degenerate Elliptic Boundary Value Problems Research Articles

Hidden convexity in the heat, linear transport, and Euler’s rigid body equations: A computational approach

A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such ‘gauge invariance’ is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by noncausal dual problems.

Bounded H^infty -calculus for a degenerate elliptic boundary value problem

On a manifold X with boundary and bounded geometry we consider a strongly elliptic second order operator A together with a degenerate boundary operator T of the form T=varphi _0gamma _0 + varphi _1gamma _1. Here gamma _0 and gamma _1 denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that varphi _0, varphi _1ge 0, and varphi _0+varphi _1ge c, for some c>0, where either varphi _0,varphi _1in C^{infty }_b(partial X) or varphi _0=1 and varphi _1=varphi ^2 for some varphi in C^{2+tau }(partial X), tau >0. We also assume that the highest order coefficients of A belong to C^tau (X) and the lower order coefficients are in L_infty (X). We show that the L_p(X)-realization of A with respect to the boundary operator T has a bounded H^infty -calculus. We then obtain the unique solvability of the associated boundary value problem in adapted spaces. As an application, we show the short time existence of solutions to the porous medium equation.

Nonuniqueness for a fully nonlinear, degenerate elliptic boundary-value problem in conformal geometry

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing σk-curvature in the interior and constant Hk-curvature on the boundary. When restricting to the closure of the positive k-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove nonuniqueness for the boundary-value problem σ4-curvature equals zero and constant H4-curvature by using bifurcation results proven by Case, Moreira and Wang. Surprisingly, our construction via products of sphere and hyperbolic space only works for a finite set of dimensions.

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing $\sigma_k$-curvature in the interior and constant $H_k$-curvature on the boundary. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem which allows us to construct examples of compact Riemannian manifolds $(X,g)$ for which this problem admits multiple non-homothetic solutions in the case when $2k<\dim X$. Our examples are all such that the boundary with its induced metric is a Riemannian product of a round sphere with an Einstein manifold.

A strong maximum principle for globally hypoelliptic operators

In this paper we study the strong maximum principle for globally hypoelliptic differential operators of second-order, and reveal the underlying analytical mechanism of propagation of maximums in terms of the Lie algebra generated by diffusion vector fields and the Fichera function. Our formulation of the strong maximum principle is coordinate-free. The results here may be applied to questions of uniqueness for degenerate elliptic boundary value problems on a manifold. Furthermore, the mechanism of propagation of maximums plays an important role in the interpretation and study of Markov processes from the viewpoint of functional analysis.

Multiple solutions for semilinear cone elliptic equations without Ambrosetti–Rabinowitz condition

In this paper we establish the existence of multiple solutions for a class of semilinear cone degenerate elliptic Dirichlet boundary value problems involving subcritical nonlinearity (cone Sobolev exponent) without the Ambrosetti–Rabinowitz condition. The paper uses singular analysis to control the linear part to provide appropriate functional setting that a variation of the Moutain Pass argument can be applied.

Weighted-W^{1,p} estimates for weak solutions of degenerate and singular elliptic equations

Global weighted $L^{p}$-estimates are obtained for the gradient of solutions to a class of linear singular, degenerate elliptic Dirichlet boundary value problems over a bounded non-smooth domain. The coefficient matrix is symmetric, nonnegative definite, and both its smallest and largest eigenvalues are proportion to a weight in a Muckenhoupt class. Under a smallness condition on the mean oscillation of the coefficients with the weight and a Reifenberg flatness condition on the boundary of the domain, we establish a weighted gradient estimate for weak solutions of the equation. A class of degenerate coefficients satisfying the smallness condition is characterized. A counter example to demonstrate the necessity of the smallness condition on the coefficients is given. Our $W^{1,p}$-regularity estimates can be viewed as the Sobolev's counterpart of the H\"{o}lder's regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni in 1982.

A note on regularity of solutions for degenerate singular elliptic boundary value problems

We prove the $$C^{1,\beta }$$ -boundary regularity and a comparison principle for weak solutions of the problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u-\lambda \psi _{p}(u)=f(x)&{}\quad \text {in }\Omega , \\ u=0&{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a bounded domain in $$\mathbb {R}^{N},N>1\ $$ with smooth boundary $$\partial \Omega ,\ \ \Delta _{p}u=\mathrm{div}(|\nabla u|^{p-2}\nabla u),\psi _{p}(u)=|u|^{p-2}u,p>1,\ $$ and f is allowed to be unbounded.

Bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws

This paper is devoted to the study of semilinear degenerate elliptic boundary value problems arising in combustion theory that obey a general Arrhenius equation and a general Newton law of heat exchange. Our degenerate boundary conditions include as particular cases the isothermal condition (Dirichlet condition) and the adiabatic condition (Neumann condition). We prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless rate of heat production. More precisely, we give sufficient conditions for our semilinear boundary value problems to have three positive solutions, which suggests that the bifurcation curves are S-shaped.

Fractional conformal Laplacians and fractional Yamabe problems

Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type maximum principle together with interplays between analysis of weighted trace Sobolev inequalities and conformal structure of the underlying manifolds, which extend the phenomena displayed in the classic Yamabe problem and boundary Yamabe problem.

Multiple solutions of semilinear degenerate elliptic boundary value problems II

AbstractThe purpose of this paper is to study a class of semilinear degenerate elliptic boundary value problems depending on a parameter which include as particular cases the Dirichlet problem and the Robin problem. By using Schauder's fixed point theorem and the Leray–Schauder degree, we derive lower bounds on the number of solutions of our problem. The results here extend earlier theorems due to Kazdan–Warner and also Amann–Hess to the degenerate case.

Degenerate elliptic boundary value problems with asymptotically linear nonlinearity

The purpose of this paper is to study a class of semilinear degenerate elliptic boundary value problems with asymptotically linear nonlinearity which include as particular cases the Dirichlet and Robin problems. Our approach is based on the global inversion theorems between Banach spaces, and is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of the variational method, we prove existence and uniqueness theorems for our problem. The results here extend three earlier theorems due to Ambrosetti and Prodi to the degenerate case.

Multiple solutions of semilinear degenerate elliptic boundary value problems II

Multiple solutions of semilinear degenerate elliptic boundary value problems II

Multiple solutions of semilinear degenerate elliptic boundary value problems

In the paper the e-mail address is incorrect. The correct one is the following: [email protected]

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 Lp 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

Symmetry of Solutions to Variational Problems for Nonlinear Elliptic Equations via Reflection Methods

We discuss some recent results on symmetry of solutions of nonlinear partial differential equations. We focus on elliptic and degenerate elliptic boundary value problems of second order with variational structure and the simple looking case where the underlying domain is radially symmetric. In this setting, we study solutions which are given as minimizers of constrained minimization problems or have low Morse index, and we examine which amount of symmetry of the data is inherited by these solutions. We highlight how the answer to this general question depends on specific assumptions on the data. The underlying techniques collected in this survey are elementary as they solely rely on hyperplane reflections and well known analytical and topological tools, but they yield surprisingly general results in situations where classical methods do not apply.

Semilinear degenerate elliptic boundary value problems at resonance

The purpose of this paper is to study a class of semilinear degenerate elliptic boundary value problems at resonance which include as particular cases the Dirichlet and Robin problems. The approach here is based on the global inversion theorems between Banach spaces, and is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of the Lyapunov–Schmidt procedure and the global inversion theorem, we prove existence and uniqueness theorems for our problem. The results here extend an earlier theorem due to Landesman and Lazer to the degenerate case.

An anti-maximum principle for degenerate elliptic boundary value problems with indefinite weights

This article deals with an anti-maximum principle to the degenerate case and to elliptic boundary value problems with indefinite weight function which include the Dirichlet and Robin boundary conditions as special cases. The result extends an earlier theorem by Clément and Peletier [An anti-maximum principle for second-order elliptic operators, J. Diff. Eqns. 34 (1979), pp. 218–229] to the degenerate case.

Degenerate elliptic boundary value problems with asymmetric nonlinearity

This paper is devoted to the study of a class of semilinear degenerate elliptic boundary value problems with asymmetric nonlinearity which include as particular cases the Dirichlet and Robin problems. The most essential point is how to generalize the classical variational approach to eigenvalue problems with an indefinite weight to the degenerate case. The variational approach here is based on the theory of fractional powers of analytic semigroups. By making use of global inversion theorems with singularities between Banach spaces, we prove very exact results on the number of solutions of our problem. The results extend an earlier theorem due to Ambrosetti and Prodi to the degenerate case.

On a semilinear degenerate elliptic boundary value problem for pseudodifferential equations

1. In this article a semilinear boundary value problem is studied for a degenerate elliptic pseudodifferential equation in the Sobolev type spaces. We prove the existence of a solution using the Borsuk theorem on topological degree (see [2]) instead of the Shauder theorem usually applied in this theory (see [1]). 2. Let 1 ≤ p < + ∞ , m ∈ � , δ = ( δ 1 , δ 2 , …, δ n – 1 , 0), δ i ≥ 0, γ i = 1 + δ i , ξ = ( ξ 1 , ξ 2 , …, ξ n ) = ( ξ ', ξ n ) ∈ � n , x = ( x 1 , x 2 , …, x n – 1 , x n ) = ( x ', x n ) ∈ � n . Introduce the notations: