We show that a piecewise linear finite element approximation of the obstacle problem gives an approximate free boundary converges, in an appropriate distance, to the free boundary of the continuous problemunder a stability condition on the obstacle.

of thesis The thesis deals with the upscaling of reactive flows in complex geometry. The reactions (deposition/dissolution) take place at a part of the boundary and depending on the size of the reaction domain, the changes in the pore structure that are due to the deposition process may or may not be neglected. In mathematical terms, the models are defined in a fixed, respectively variable geom...

Scale-free surfaces, such as cones, remain unchanged under a simultaneous expansion of all coordinates by the same factor. Probability density of a particle diffusing near such absorbing surface at large time approaches a simple form that incorporates power-law dependencies on time and distance from a special point, such as apex of the cone, which are characterized by a single exponent η. The s...

Free boundaries arise as the interface betweenmaterials inwhich thematerials retain some energy. In contrast, the interface represented by a minimal surface lives in an ambient space that is empty. Despite this difference between these two types of interfaces, there are deep connections between them. In a ground-breaking paper in 1980, Alt and Caffarelli showed that strategies and tools from th...

For a bounded domain D ⊂ Rn, we study minimizers of the energy functional ∫ D |∇u| dx+ ∫ D∩(Rn−1×{0}) λχ{u>0} + λ χ{u<0} dHn−1, without any sign restriction on the function u. One of the main result states that the free boundaries Γ = ∂{u(·, 0) > 0} and Γ− = ∂{u(·, 0) < 0} never touch. Moreover, using Alexandrov-type reflection technique, we can show that in dimension n = 3 the free boundaries ...

This paper concentrates on the propagation of waves in a layer of binary mixture of elastic solids subjected to stress free boundaries. Secular equations for the layer corresponding to symmetric and antisymmetric wave modes are derived in completely separate terms. The amplitudes of displacement components and specific loss for both symmetric and antisymmetric modes are obtained. The effect of ...

There is by now a significant literature on homological stability properties of various families of discrete groups, including matrix groups, mapping class groups of surfaces, and automorphism groups of free groups. The present paper is yet another contribution in this direction, for a family of groups An,k that can be thought of as relative versions of automorphism groups of free groups. These...

We propose a mathematical model for the process of dry pasta cooking with specific reference to spaghetti. Pasta cooking is a two-stage process: water penetration followed by starch gelatinization. Differently from the approach adopted so far in the technical literature, our model includes free boundaries: the water penetration front and the gelatinization onset front representing a fast stage ...

This paper deals with a free boundary problem for both Laplacian and p-Laplacian operators. We begin by proving the existence of solution (which is of class C) for the associated shape optimization problem. Then, after performing the shape derivative we will present two approaches in order to get sufficient conditions of existence of the free boundaries. The first one needs the use of some maxi...