Maria Agostina Vivaldi

Harnack inequalities for fuchsian type weighted elliptic equations

Harnack type inequalities for nonnegative (weak) solutions of degenerate elliptic equations, in divergence form, are established. The asymptotic behavior of solutions of Fuchsian type weighted elliptic operators is also investigated.

On the Laplacean transfer across fractal mixtures

Laplacean transport across and towards irregular interfaces have been used to model many phenomena in nature and technology. The peculiar aspect is that these phenomena take place in domains with small bulk and large interfaces in order to produce rapid and efficient transport. In this paper we perform the asymptotic homogenization analysis of Robin problems in domains with a fractal boundary.

Bilateral inequalities and implicit unilateral systems of the non-variational type

We prove that ageneralized solution exists for a bilateral problem relative to second order linear elliptic operator with principal part not in divergence form. We also prove that this solution isregular, when the obstacles are sufficiently regular, and that it provides an appropriate substitute for the solution when the hypotheses of regularity are not satisfied. A probabilistic characterization of this solution is also given.

EXISTENCE AND REGULARITY RESULTS FOR OBLIQUE DERIVATIVE PROBLEMS FOR HEAT EQUATIONS IN AN ANGLE

An initial-boundary-value problem is considered for the heat equation in an infinite angle d θr ⊆ R 2 × [0, ∞) with the oblique derivative boundary conditions on the faces λ i of the angle: with either h 0 + h 1 > 0, or h 0 + h 1 ≦ 0. The unique solvability of such a problem is proved in appropriate weighted Sobolev spaces according to the sign of h 0 + h 1 . Estimates of the solution are obtained under ‘natural’ restrictions on the opening of the angle.

Bilateral evolution problems of non-variational type: Existence, uniqueness, Hölder-regularity and approximation of solutions

We study bilateral problems for a second order parabolic operator with principal part not of divergence form. We prove existence, uniqueness, Holder-continuity and approximation results for both strong and generalized solutions.

Stability of free boundaries

IN THIS paper we consider a unilateral problem relative to a nonlinear elliptic operator A-G of the second order. A is a linear operator whose principal part is not in divergence form and G is a Nemytsky operator relative to a continuous function with quadratic growth in the gradient. We introduce unilateral problems that are “regularizing”, relative to a sequence of operators A” G” which converge to A G. The An’s have regular coefficients; the G”‘s have linear growth in the gradient. The operators A” can obviously be written in divergence form. In this paper we show that the free boundaries of the “regular” problems converge to the free boundary of the “nonregular” problem; the rate of convergence can be expressed in terms of the convergence of the operators. The convergence in the uniform norm of the solutions was proved in a more general framework by the present authors in [6]. In this paper an exact estimate of the rate of convergence of the solutions in terms of the convergence of the coefficients of the operators is also obtained, (for the linear case see also Troianiello [S]). This kind of estimate will be used in the present paper. Caffarelli [3] and Friedman [5] introduce an assumption of “sign” and of regularity on the data in order to study the “regularity” of the free boundary of a variational inequality relative to the Laplacian and with zero obstacle. Under this assumption they prove a maximum principle in the noncoincidence set, together with a property of “nondegeneracy” (i.e. the solution leaves the obstacle with a certain minimum speed). Brezzi and Caffarelli consider the dam problem and give an estimate of the rate of convergence of the “discrete” free boundariesrelative to the approximation with affine finite elements-to the “continuous” free boundary of the dam problem [2]. They make use of the regularity properties of the solution Wi and of the free boundary, of the property of nondegeneracy proved in [3] and [5] and of the estimate of the convergence of the discrete solutions in the uniform norm. The question we deal with in the present paper is, in a sense, opposite to that considered by Brezzi and Caffarelli; what we want to do is to approximate a problem having a “nonregular” solution by means of “regularizing” problems. We take up the hypothesis of the sign on the data again, and prove a maximum principle for the solutions, uniform in n, together

WEIGHTED ESTIMATES ON FRACTAL DOMAINS

The aim of the paper is to establish estimates in weighted Sobolev spaces for the solutions of the Dirichlet problems on snowflake domains, as well as uniform estimates for the solutions of the Dirichlet problems on pre-fractal approximating domains. §

Approximation results for bilateral nonlinear problems of nonvariational type

IN THIS paper we give a few approximation results for bilateral evolution problems for linear and quasi-linear second order operators whose principal part is not in divergence form. Existence and uniqueness theorems for strong solutions of the above mentioned problems are given in [l] and in [2]. In the first section we consider the case of the linear operator E; suitable “regularizing” operators E” are introduced, whose coefficients converge to the coefficients of E. Due to the regularity hypothesis, the operators E” can be written in divergence form. We study the order of convergence of the solutions of the regularizing problems to the solution of the given problem. This is done in the uniform norm and in terms of the order of convergence of the coefficients. We observe that these results cannot be obtained directly from the approximation results for the equations by using the Lewy-Stampacchia inequality. In the second section we study bilateral problems for the quasi-linear operator E G, where G is a Nemytsky operator associated with a continuous function g(x, t, U, p) with quadratic growth in the gradient variable p. We introduce suitable regularizing operators E” G” which converge to the operator E G and results similar to those of the linear case are obtained. Finally, in Section 3 we give a few examples of operators G” converging to G. As is well known only the problems with “very regular” obstacles admit strong solutions. (See [31, [41, [l] and [j]). I n p revious papers [l] and [j] we have introduced, in the linear case, a definition of generalized solution which provides an appropriate substitute for the solution when the latter does not exist. This is the case, for instance, when the obstacles are only continuous. These approximation results can be extended to the case of generalized solutions in the case of Holder-regular obstacles, see [l].

Dynamical Quasi-Filling Fractal Layers

The aim of this paper is to investigate second order transmission problems across quasi-filling dynamical layers from the point of view of the variational convergence of energy forms. We prove that the solution to the second order transmission problem across a Koch-type curve is the limit of the solutions to suitable second order transmission problems across polygonal curves.

Regularity results for p-Laplacians in pre-fractal domains

Abstract We study obstacle problems involving p-Laplace-type operators in non-convex polygons. We establish regularity results in terms of weighted Sobolev spaces. As applications, we obtain estimates for the FEM approximation for obstacle problems in pre-fractal Koch Islands.