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Phase transition problems with the line tension effect: the super-quadratic case
No full textLet be an open bounded set of and let and be two non-negative continuous functions vanishing at and , respectively. We analyze the asymptotic behavior as , in terms of -convergence, of the following functional
where is a scalar density function and denotes its trace on . We show that the singular limit of the energies leads to a coupled problem of bulk and surface phase transitions
Subcritical approximation of the Sobolev quotient and a related concentration result
No full textLet be a general, possibly non-smooth, bounded domain of , . Let be the critical Sobolev exponent. We study the following variational problem
investigating its asymptotic behavior as goes to zero, by means of \gamp-convergence techniques.
We also show that sequences of maximizers concentrate energy at one point
The Dirichlet problem for the p-fractional Laplace equation
No full textWe deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order s in (0,1) and summability growth p in (1,infty), whose model is the fractional p-Laplacian operator with measurable coefficients. We review several recent results for the corresponding weak solutions/supersolutions, as comparison principles, a priori bounds, lower semicontinuity, boundedness, Hölder continuity up to the boundary, and many others. We then discuss the good definition of (s,p)-superharmonic functions, and the nonlocal counterpart of the Perron method in nonlinear Potential Theory, together with various related results. We briefly mention some basic results for the obstacle problem for nonlinear integro-differential equations. Finally, we present the connection amongst the fractional viscosity solutions, the weak solutions and the aforementioned (s,p)-superharmonic functions, together with other important results for this class of equations when involving general measure data, and a surprising fractional version of the Gehring lemma.
We sketch the corresponding proofs of some of the results presented here, by especially underlining the development of new fractional localization techniques and other recent tools. Various open problems are listed throughout the paper
Developments and perspectives in Nonlinear Potential Theory
Full text linkNonlinear Potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered. In recent years there has been a substantial development of this subject, mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials. Here we give a brief account of such developments and outline the connections with different fields
Γ-Convergence of some super quadratic functionals with singular weights
No full textWe study the -convergence of the following functional ()
where is an open bounded set of and and are two non-negative continuous functions vanishing at and , respectively.
In the previous functional, we fix and is a scalar density function, denotes its trace on , stands for the distance function to the boundary \partial\Om.
We show that the singular limit of the energies leads to a coupled problem of bulk and surface phase transitions
Gamma-convergence for one-dimensional nonlocal phase transition energies
Full text linkWe study the asymptotic behavior as ε goes to 0 of an appropriate scaling of the following nonlocal Allen-Cahn energy,where I is an interval in R, and W is a double-well potential. We provide a Γ-convergence result for any s ∈ (0,1), by extending the case when s=1/2 studied by Alberti, Bouchittè and Seppecher in [2]. We also investigate the convergence as s↗1 of the related optimal profile problem to the local counterpart
Nonlinear parabolic problems with lower order terms and related integral estimates
No full textWe deal with the solutions to nonlinear parabolic equations of the form
under standard growth conditions on and , with only assumed to be integrable to the power . We prove general local decay estimates for level sets of the solutions and the gradient which imply very general estimates in rearrangement function spaces (Lebesgue, Orlicz, Lorentz) and non-rearrangement ones, up to Lorentz–Morrey spaces
Fractional Regularity for Nonlinear Elliptic Problems with Measure Data
No full textWe consider nonlinear elliptic equations of the type having a Radon measure on the right-hand side and prove fractional differentiability results of Calder\'on-Zygmund type for very weak solutions. We extend some of the results achieved by G. Mingione (Ann. Scu. Norm. Sup., 2007), in turn improving a regularity result by Cirmi \& Leonardi (DCDS-A, 2010)
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