A priori bounds for a class of semi-linear degenerate elliptic equations

In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a ij(x)■ij u+b i(x)■iu+f(x,u)=0,in ΩRn,(*)where aij■iφ■jφ=0 on■Ω,andφis the defining function of ■Ω.Imposing suitable conditions on the coefficients and f(x,u),one can get the L∞-estimates of(*)via blow up method.     

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.      

A priori estimate for non-uniform elliptic equations

A priori estimate for non-uniform elliptic equations with periodic boundary conditions is concerned. The domain considered consists of two sub-regions, a connected high permeability region and a disconnected matrix block region with low permeability. Let ? denote the size ratio of one matrix block to the whole domain. It is shown that in the connected high permeability sub-region, the Hölder and the Lipschitz estimates of the non-uniform elliptic solutions are bounded uniformly in ?. But Hölder gradient estimate and Lp estimate of the second order derivatives of the solutions in general are not bounded uniformly in ?.     

Potential bounds in problems on quasilinear elliptic equations

We study quasilinear elliptic equations with strong nonlinear terms and systems of such equations. The methods developed by the authors in [1], [2] are used to prove the existence of solutions for boundary—value problems using some information on behavior of potential bounds for nonlinearities; the L–characteristics of elliptic operators and their fractional powers play an important role. New conditions are suggested for the existence of classical solutions of quasilinear second order elliptic equations.     

A new maximum principle of elliptic differential equations in divergence form

In this paper will be presented a new maximum principle of elliptic differential equations in divergence form which can be regarded as the counterpart of the Alexandroff-Bakelman-Pucci maximum principle of elliptic differential equations in nondivergence form.

     

Near operators theory and fully nonlinear elliptic equations

We give a short survey of the Campanato near operators theory and of its applications to fully nonlinear elliptic equations.     

Existence results and a priori bounds for higher order elliptic equations and systems

We apply degree theory to prove the existence of positive solutions of semilinear elliptic systems. As an application we obtain a number of new results for higher order equations which appear frequently in applications. In particular, we extend to these equations and systems the notions of sublinearity and superlinearity, classical in the setting of second order equations.     

A priori bounds on difference quotients of solutions to some linear uniformly elliptic difference equations

Second order difference quotients of solutions to a class of linear uniformly elliptic difference Dirichlet problems are bounded in terms of quantities which depend on the coefficients of the operator, the inhomogenous term, the boundary values and the domain-which we take to be a rectangle. The results we obtain have theoretical and practical applications.     

A priori bounds in the cauchy problem for coupled elliptic systems

By means of the logarithmic convexity of a suitable functional, an a priori inequality is developed for the sum of the squares of the solutions of the following improperly posed Cauchy problem. Consider the coupled elliptic system Lu = aν+f,Lν= bu+g, where L is a uniformly elliptic differential operator, a,b,f and g are bounded integrable functions with |b(x)|≧b₀>0 and ν satisfies a stabilizing condition, and where upper bounds for the error in measurement of the Cauchy data on the initial surface are prescribed. From the a priori estimate uniqueness, stability, and pointwise bounds for the solutions u and n are simultaneously deduced. The bounds are improvable by the Ritz technique. Moreover, the method presented here can be extended to the nonlinear system Lu = f(x, ν), Lν =g(x,u)provided g is a suitable form     

Inequalities and bounds for elliptic integrals

Computable lower and upper bounds for the symmetric elliptic integrals and for Legendre's incomplete integral of the first kind are obtained. New bounds are sharper than those known earlier. Several inequalities involving integrals under discussion are derived.     

A higher integrability result for nondivergence elliptic equations

The aim of this paper is to establish a higher integrability result of the second derivatives of solutions to nondivergence elliptic equations of the type . We assume that the coefficients a _ij are bounded and have small BMO-norm.      

Maximum principles and bounds for a class of fourth order nonlinear elliptic equations

We deduce maximum principles for a class of fourth order nonlinear elliptic equations by using auxiliary functions containing the square of the second gradient of the solution of such equations. A priori bounds on various quantities of interest are obtained.     

A priori bounds for an indefinite superlinear elliptic equation with exponential growth

This paper considers positive solutions to problem

     

A priori estimates for a class of degenerate elliptic equations

In this paper we investigate the regularity of solutions for the following degenerate partial differential equation $$\left \{\begin{array}{ll} -\Delta_p u + u = f \qquad {\rm in} \,\Omega,\\ \frac{\partial u}{\partial \nu} = 0 \qquad \qquad \,\,\,\,\,\,\,\,\,\, {\rm on} \,\partial \Omega, \end{array}\right.$$ when ${f \in L^q(\Omega), p > 2}$ and q ≥ 2. If u is a weak solution in ${W^{1, p}(\Omega)}$ , we obtain estimates for u in the Nikolskii space ${\mathcal{N}^{1+2/r,r}(\Omega)}$ , where r = q(p ? 2) + 2, in terms of the L ^q norm of f. In particular, due to imbedding theorems of Nikolskii spaces into Sobolev spaces, we conclude that ${\|u\|^r_{W^{1 + 2/r - \epsilon, r}(\Omega)} \leq C(\|f\|_{L^q(\Omega)}^q + \| f\|^{r}_{L^q(\Omega)} + \|f\|^{2r/p}_{L^q(\Omega)})}$ for every ${\epsilon > 0}$ sufficiently small. Moreover, we prove that the resolvent operator is continuous and compact in ${W^{1,r}(\Omega)}$ .     

A priori estimates for solutions of nonlinear second-order elliptic equations

We consider classes of elliptic equations of the form (x,u,u D ² u)=0 for the solutions of which one establishes local and global a priori estimates for D ² u=. In particular, one investigates the Monge-Ampere equation, and for its convex solutions one constructs a local and a global estimate for D ² u and a local estimate for.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 59, pp. 31–59, 1976.     

Liouville-type results for semilinear elliptic equations in unbounded domains

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space:

Implicit elliptic differential equations

Let be a bounded domain in ⁿ (n 3) having a smooth boundary, letY be a closed, connected and locally connected subset of ^h , letf be a real-valued function defined on × ^h × ^nh ×Y, and letL be a linear, second-order elliptic operator. In this paper, the existence of strong solutionsu W ^2,p (, ^h ) W ₀ ^1,p (, ^h ) (n<p<+) to the implicit elliptic equationf(x, u, Du, Lu)=0, whereu=(u ₁,u ₂, ...,u _h ),Du=(Du ₁,Du ₂, ...,Du _h ) andLu=(Lu ₁,Lu ₂, ...,Lu _h ), is established. The abstract framework where the equation is studied is that of set-valued analysis.Dedicated to Professor G. Pulvirenti on the occasion of his sixtieth birthday