1,721,012 research outputs found
Geometric inequalities and symmetry results for elliptic systems
No full textWe obtain some Poincaré type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable solutions of possibly degenerate elliptic systems of the form div (a (jruj)ru) = F1(u; v); div (b (jrvj)rv) = F2(u; v); where F 2 C1;1 loc (R2). Our setting is very general, and it comprises, as a particular case, a conjec- ture of De Giorgi for phase separations in R2
Asymptotics of fractional perimeter functionals and related problems
No full textIn this paper we review some recent results concerning the asymptotics of a fractional perimeter and the regularity of the corresponding minimizers. We also provide an elementary example of set with infinite s-perimeter
Symmetry results for stable and monotone solutions to fibered systems of PDEs
No full textWe study the symmetry properties for solutions of elliptic systems of the type [eqution presnted] We obtain a Poincaré-type formula for the solutions of the system and we use it to prove a symmetry result both for stable and for monotone solutions
(Non)local and (non)linear free boundary problems
Full text linkWe discuss some recent developments in the theory of free boundary problems, as obtained in a series of papers in collaboration with L. Caffarelli, A. Karakhanyan and O. Savin. The main feature of these new free boundary problems is that they deeply take into account nonlinear energy superpositions and possibly nonlocal functionals. The nonlocal parameter interpolates between volume and perimeter functionals, and so it can be seen as a fractional counterpart of classical free boundary problems, in which the bulk energy presents nonlocal aspects. The nonlinear term in the energy superposition takes into account the possibility of modeling different regimes in terms of different energy levels and provides a lack of scale invariance, which in turn may cause a structural instability of minimizers that may vary from one scale to another
Concentration of solutions for a singularly perturbed mixed problem in non-smooth domains
No full textAbstractWe consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain Ω⊂Rn whose boundary has an (n−2)-dimensional singularity. Assuming 1<p<n+2n−2, we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero
On a fractional harmonic replacement
Full text linkGiven s ∈ (0, 1), we consider the problem of minimizing the fractional Gagliardo seminorm in Hs with prescribed condition outside the ball and under the further constraint of attaining zero value in a given set K. We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set A to K increases the energy of at most the measure of A (this may be seen as a perturbation result for small sets A). Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions
Continuity and density results for a one-phase nonlocal free boundary problem
Full text linkWe consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are Hölder continuous and the free boundary has positive density from both sides.
For this, we also introduce a new notion of fractional harmonic replacement in the extended variables and we study its basic properties
Boggio's formula for fractional polyharmonic Dirichlet problems
Full text linkBoggio's formula in balls is known for integer-polyharmonic Dirichlet problems and for fractional Dirichlet problems with fractional parameter less than 1. We give here a consistent formulation for fractional polyharmonic Dirichlet problems such that Boggio's formula in balls yields solutions also for the general fractional case
A density property for fractional weighted Sobolev spaces
Full text linkIn this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant
Isolamento e caratterizzazione di un gene codificante per una acquaporina del tonoplasto in radici di zucca (Cucurbita pepo L.). Studi preliminari
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