Degenerate Elliptic Problems Research Articles

On the refined maximum principle for degenerate elliptic and parabolic problems

We extend the refined maximum principle in [H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47–92] to degenerate elliptic and parabolic equations with unbounded coefficients. Then we discuss the well-posedness of the corresponding Dirichlet boundary value problems.

Asymptotic behaviour of solutions for a class of degenerate elliptic critical problems

We study the asymptotic behaviour of finite energy solutions to subelliptic equations with critical nonlinearity for the operator L = Δ x + | x | 2 α Δ y , α > 0 . In particular, we obtain the exact behaviour at infinity of the related Sobolev extremals.

Criteria for well-posedness of degenerate elliptic and parabolic problems

We provide explicit criteria for uniqueness or nonuniqueness of solutions to a wide class of second order elliptic and parabolic problems. The operator coefficients may be unbounded or vanish, or not to have a limit when approaching some part of the boundary, referred to as singular boundary. We discuss whether boundary conditions should be imposed on such a part to ensure well-posedness. The answer depends on the dimension of the singular boundary, and possibly on the behavior of coefficients near it.

Degenerate Elliptic Eigenvalue Problems with Indefinite Weights

The purpose of this paper is to provide a careful and accessible exposition of the Kreĭn and Rutman Theory of degenerate elliptic eigenvalue problems with indefinite weights that model population dynamics in environments with spatial heterogeneity. We prove that the first eigenvalue of our problem is algebraically simple and its corresponding eigenfunction may be chosen to be positive everywhere. Here the approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. The results extend an earlier theorem due to Manes and Micheletti to the degenerate case.

Bifurcation in weighted Sobolev spaces

When P(x, ∂) is a second order linear elliptic differential operator on many bifurcation problems P(x, ∂)u − λu + f(x, u) = 0 cannot be formulated as a functional equation from to irrespective of p ∊ [1, ∞], either because the Nemystskii operator f (u) ≔ f(x, u) does not map W2,p to Lp due to the growth of f as |x| → ∞ or because, while well defined, f is not Fréchet differentiable. Far from being pathological, the latter may happen even when f is C∞. In this paper, we show that all these difficulties may often be circumvented by replacing the spaces W2,p and Lp by weighted spaces and where ω is an ‘admissible’ weight and p ∊ (1, ∞), p > N/2. Even though the admissibility of ω depends in part upon f, this still yields a bifurcation theorem in W2,p due to the inclusion . In addition, this approach can be fine tuned to discuss bifurcation in some degenerate elliptic problems after a suitable change of the variables x and u. The problem −|x|4Δu + (Q(x) − λ)u − g(u) = 0 is treated as an example.

The inverse mean curvature flow as an obstacle problem

The inverse mean curvature flow is a geometric evolution problem that is turned into a degenerate elliptic problem by a level set formulation. In the latter form, it may be regarded as a special case of an obstacle problem. In this paper, solutions of the problem are constructed with an approximation by p-harmonic functions and the use of appropriate barrier functions. This also extends the known existence results for weak solutions of the inverse mean curvature flow.

Degenerate elliptic boundary-value problems of second order in nonsmooth domains

Degenerate elliptic boundary-value problems of second order in nonsmooth domains

The influence of domain geometry in the boundary behavior of large solutions of degenerate elliptic problems

In this paper we study the asymptotic boundary behavior of large solutions of the equation \Delta u=d^{\alpha}u^p in a regular bounded domain \Omega in \R^N , N\geq 2 , where d(x) denotes the distance from x to \partial\Omega , p>1 and \alpha>0 . We precise the expansion which depends on the mean curvature of the boundary.

Bifurcation points of a degenerate elliptic boundary-value problem

We consider the nonlinear elliptic eigenvalue problem \begin{align*} -\nabla\cdot\{A(x)\nabla u(x)\} & =\lambda f(u(x))\text{ for }x\in\Omega\\ u(x) & =0\text{ for }x\in\partial\Omega \end{align*} where \Omega is a bounded open subset of \mathbb{R}^{N} and f\in C^{1}(\mathbb{R}) with f(0)=0 and f^{\prime}(0)=1 . The ellipticity is degenerate in the sense that 0\in\Omega and A(x)>0 for x\neq0 but \lim_{x\rightarrow0}\frac{A(x)}{\left\vert x\right\vert ^{2}}=1 . We show that there is vertical bifurcation at all points \lambda in the interval (\frac{N^{2}}{4},\infty). Bifurcation also occurs at any eigenvalues of the linearized problem that are below \frac{N^{2}}{4} . Our treatment is based on recent results concerning the bifurcation points of equations with nonlinearities that are Hadamard differentiable, but not Fréchet differentiable.

Renormalized solutions of degenerate elliptic problems

We consider a general class of degenerate elliptic problems of the form A u + g ( x , u , D u ) = f , where A is a Leray–Lions operator from a weighted Sobolev space into its dual. We assume that g ( x , s , ξ ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L 1 -setting.

A sharp estimate of the extinction time for the mean curvature flow

We establish a pointwise comparison result for a nonlinear degenerate elliptic Dirichlet problem using an isoperimetric inequality involving the total mean curvature. In particular this result provides a sharp estimate for the extinction time of a class of compact surfaces, wider than the convex one, moving by mean curvature flow. Finally we present numerical experiments to compare our estimate with those known in literature.

Flatness of Bernoulli jets

Level sets of minimizers of a possibly singular or degenerate elliptic variational problem related with fluid jets are shown to possess a Harnack-type inequality. A few flatness and one-dimensional results thence follow.

Multiresolution weighted norm equivalences and applications

We establish multiresolution norm equivalences in weighted spaces L2w((0,1)) with possibly singular weight functions w(x)≥0 in (0,1). Our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions. The discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function w(x) within each scale. Since norm equivalences for Sobolev norms are by now well-known, our result can also be applied to weighted Sobolev norms. We apply our theory to the problem of preconditioning p-Version FEM and wavelet discretizations of degenerate elliptic and parabolic problems from finance.

Quasi-linear degenerate elliptic problems with L1L1 data

Quasi-linear degenerate elliptic problems with L1L1 data

Schauder estimates for degenerate elliptic and parabolic problems with unbounded coefficients in ${\Bbb R}^N$

We consider a class of second-order degenerate elliptic operators. Continuing the study started in [3], we prove Schauder estimates for the distributional solutions of the nonhomogeneous elliptic equation $\lambda u-{\mathcal A}u=f$ and the Cauchy problem $D_tu={\mathcal A}u+g$, $u(0,\cdot)=f$.

Regularity for degenerate elliptic problems via p-harmonic approximation

We introduce a new method to prove regularity of solutions to certain degenerate elliptic problems. The method is based on the p-harmonic approximation lemma, recently proved by the authors in [F. Duzaar, G. Mingione, The p-harmonic approximation and the regularity of p-harmonic maps, Calc. Var., 2004, in press], that allows to approximate functions with p-harmonic functions in the same way as the classical harmonic approximation lemma (going back to De Giorgi) does via harmonic functions. The method presented here also bypasses certain difficulties arising when treating some degenerate and singular problems with a weak structure, such as degenerate and singular quasiconvex integrals, and provides transparent and elementary proofs.

A class of quasilinear degenerate elliptic problems

We establish the existence of solutions for a class of quasilinear degenerate elliptic equations. The equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain, and degeneracy only on the boundary. Equations of transonic gas dynamics, for example, satisfy this property in the region of subsonic flow and are degenerate across the sonic surface. We prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary. The maximal rate of blow-up at the degenerate boundary is bounded by the “degree of degeneracy” in the principal coefficients of the quasilinear elliptic operator. Our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation, the pressure-gradient equations of the compressible Euler equations, and the singular quasilinear anisotropic elliptic problems, and extend to the class of equations which satisfy the structure condition, such as the shallow water equation, compressible isentropic two-dimensional Euler equations, and general two-dimensional nonlinear wave equations. Our study provides a general framework to analyze degenerate elliptic problems arising in the self-similar reduction of a broad class of two-dimensional Cauchy problems.

Semilinear elliptic boundary-value problems in combustion theory

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. 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.

Bifurcation phenomena associated to a class of p -Laplacian like operators

Global bifurcation of positive solutions for some degenerate quasilinear elliptic problems is considered. The uniform estimate of the gradient of weak solutions is given. This estimate is crucial in our arguments.

Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on $ {\Bbb R}^n $

Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on $ {\Bbb R}^n $