Regularity of solutions of a degenerate elliptic variational problem

Summary We prove regularity of minimizers of the functional recently suggested by Ericksen [10] for the statics of nematic liquid crystals. We show that, given locally minimizing pairs (s, u),s has a continuous representative, ands, u are smooth outside the set {s=0}. The proof relies upon higher integrability estimates, monotonicity, and decay lemmas.     

On the Dirichlet-type problem for elliptic systems degenerate at a line

In this paper, the Dirichlet-type problem for the system of elliptic equations of second order with the degeneracy at a line crossing the domain is considered. The Dirichlet-type problem with additionally given asymptotics of the solution at this line is discussed. The uniqueness and the existence of the solution of this problem in the class of Hölder functions is proved.     

On a variational problem for radial solutions to extremal elliptic equations

On the ball |x| ≤ 1 of R ^m , m ≥ 2, a radial variational problem, related to a priori estimates for solutions to extremal elliptic equations with fixed ellipticity constant α is investigated. Such a problem has been studied and solved [see Manselli Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] in L ^p spaces, with p ≤ m. In this paper, we assume p > m and we prove the existence of a positive number α ₀ = α ₀(p,m) such that if there exists a smooth function maximizing the problem, whose representation is explicitly determined as in Manselli [Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] This fact is no longer true if 0 < α < α ₀.      

Mixed finite element approximation of a degenerate elliptic problem

Summary. We present a mixed finite element approximation of an elliptic problem with degenerate coefficients, arising in the study of the electromagnetic field in a resonant structure with cylindrical symmetry. Optimal error bounds are derived. Received May 4, 1994 / Revised version received September 27, 1994     

The submartingale problem for a class of degenerate elliptic operators

We consider the degenerate elliptic operator acting on ${C^2_b}$ functions on [0,∞) ^d : $$\mathcal{L}f(x)=\sum_{i=1}^d a_i(x) x_i^{\alpha_i} \frac{\partial^2 f}{\partial x_i^2} (x) +\sum_{i=1}^d b_i(x) \frac{\partial f}{\partial x_i}(x), $$ where the a _i are continuous functions that are bounded above and below by positive constants, the b _i are bounded and measurable, and the ${\alpha_i\in (0,1)}$ . We impose Neumann boundary conditions on the boundary of [0,∞) ^d . There will not be uniqueness for the submartingale problem corresponding to ${\mathcal{L}}$ . If we consider, however, only those solutions to the submartingale problem for which the process spends 0 time on the boundary, then existence and uniqueness for the submartingale problem for ${\mathcal{L}}$ holds within this class. Our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations $$ {\rm d}X_t^i=\sqrt{2a_i(X_t)} (X_t^i)^{\alpha_i/2}{\rm d}W^i_t + b_i(X_t) {\rm d}t + {\rm d}L_t^{X^i},\quad X^i_t \geq 0, $$ where ${W_t^i}$ are independent Brownian motions and ${L^{X_i}_t}$ is a local time at 0 for X ⁱ .     

On a class of degenerate potential elliptic system

We study a class of degenerate potential elliptic systems of the form , defined on a bounded or unbounded domain.     

On some degenerate evolution variational inequalities

Summary A class of parabolic variational inequalities is investigated, where: the unilateral constraints concern the time derivative of the unknown function; degeneracies or singularities with respect to the time variable are considered. Some general abstract existence and uniqueness results are proved, in the natural framework of suitable weighted spaces.

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Sunto Si studia una classe di disequazioni variazionali paraboliche, dove: i vincoli unilaterali riguardano la derivata rispetto al tempo delta funzione incognita; vengono considerate singolarità e degenerazioni rispetto alla variabile temporale. Si dimostrano alcuni risultati generali di esistenza ed unicità, nell'ambito naturale di opportuni spazi con peso.

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This work was supported in part by the «G.N.A.F.A. del C.N.R.», and by the «Ministero dell'Università e della Ricerca Scientifica» (through 60% and 40% grants).     

Existence of solutions to a general geometric elliptic variational problem

We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closed m dimensional subsets of \({\mathbf {R}}^n\) which is stable under taking smooth deformations homotopic to the identity and under local Hausdorff limits. We prove that the minimiser exists inside the class and is an \(({\mathscr {H}}^m,m)\) rectifiable set in the sense of Federer. The class of competitors encodes a notion of spanning a boundary. We admit unrectifiable and non-compact competitors and boundaries, and we make no restrictions on the dimension m and the co-dimension \(n-m\) other than \(1 \le m < n\). An important tool for the proof is a novel smooth deformation theorem. The skeleton of the proof and the main ideas follow Almgren’s (Ann Math (2) 87:321–391, 1968) paper. In the end we show that classes of sets spanning some closed set B in homological and cohomological sense satisfy our axioms.     

On a cubic variational problem

An extremal curve of the simplest variational problem is a continuously differentiable function. Hilbert’s differentiability theorem provides a sufficient condition for the existence of the second derivative of an extremal curve. It is desirable to have a simple example in which the condition of Hilbert’s theorem is violated and an extremal curve is not twice differentiable.In this paper, a cubic variational problem with the following properties is analyzed. The functional of the problem is bounded neither above nor below. There exists an extremal curve for this problem which is obtained by sewing together two different extremal curves and not twice differentiable at the sewing point. Despite this unfavorable situation, an attempt to apply the method of steepest descent (in the form proposed by V.F. Dem’yanov) to this problem is made. It turns out that the method converges to a stationary curve provided that a suitable step size rule is chosen.     

THE ELLIPTIC VARIATIONAL INEQUALITIES WITH DOUBLE DEGENERATE AND GENERAL GROWTH CONDITIONS

A class of quasilinear elliptic variatlonal inequalities with double degenerate is discussed in this paper. We extend the Keldys-Fichera boundary value problem and the first boundary problem of degenerate elliptic equation to the variatlonal inequalities. We establish the existence and uniquenees of the weak solution of conespondlng problem under monstandard growth conditions.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

On one boundary-value problem for a strongly degenerate second-order elliptic equation in an angular domain

We prove the existence and uniqueness of a classical solution of a singular elliptic boundary-value problem in an angular domain. We construct the corresponding Green function and obtain coercive estimates for the solution in the weighted Hölder classes.     

Existence and regularity of positive solutions of a degenerate elliptic problem

Existence and regularity of positive solutions of a degenerate elliptic Dirichlet problem of the form in Ω, on , where Ω is a bounded smooth domain in , , are obtained via new embeddings of some weighted Sobolev spaces with singular weights and . It is seen that and admit many singular points in Ω. The main embedding results in this paper provide some generalizations of the well‐known Caffarelli–Kohn–Nirenberg inequality.