Volume 17, Issue 2 - Special issue dedicated to Ireneo Peral. Guest editors: Antonio Ambrosetti and David Arcoya

In this article we present a survey of the Ph.D. theses that have been completed under the advice of Ireneo Peral.Following a chronological order, we summarize the main results contained in the works of the former students of Ireneo Peral.

Kamin and Vázquez [11] proved in 1991 that solutions to the Cauchy–Dirichlet problem for the porous medium equation ut=(um)x⁢x${u_{t}=(u^{m})_{xx}}$, m>1${m>1}$, on the half-line with zero boundary data and nonnegative compactly supported integrable initial data behave for large times as a dipole-type solution to the equation having the same first moment as the initial data, with an error which is o⁢(t-1/m)${o(t^{-1/m})}$. However, on sets of the form 0<x<g⁢(t)${0<x<g(t)}$, with g⁢(t)=o⁢(t1/(2⁢m))${g(t)=o(t^{1/(2m)})}$ as t→∞${t\to\infty}$, in the so-called near field, a scale which includes the particular case of compact sets, the dipole solution is o⁢(t-1/m)${o(t^{-1/m})}$, and their result gives neither the right rate of decay of the solution nor a nontrivial asymptotic profile. In this paper, we will improve the estimate for the error, showing that it is o⁢(t-(2⁢m+1)/(2⁢m2)⁢(1+x)1/m)${o(t^{-(2m+1)/(2m^{2})}(1+x)^{1/m})}$. This allows in particular to obtain a nontrivial asymptotic profile in the near field limit, which is a multiple of x1/m${x^{1/m}}$, thus improving in this scale the results of Kamin and Vázquez.

We study the behavior of the heat content for a nonlocal evolution problem.We obtain an asymptotic expansion for the heat content of a set D , defined asℍDJ⁢(t):=∫Du⁢(x,t)⁢𝑑x${\mathbb{H}_{D}^{J}(t):=\int_{D}u(x,t)\,dx}$, with u being the solution to ut=J∗u-u${u_{t}=J\ast u-u}$ withinitial condition u0=χD${u_{0}=\chi_{D}}$. This expansion is given in terms of geometric values of D . As a consequence, we obtain that ℍDJ⁢(t)=|D|-PJ⁢(D)⁢t+o⁢(t)${\mathbb{H}^{J}_{D}(t)=\lvert D\rvert-P_{J}(D)t+o(t)}$ as t↓0${t\downarrow 0}$.We also recover the usual heat content for the heat equation when we rescale the kernel J in an appropriate way.Finally, we also find an asymptotic expansion for the nonlocal analogous to the spectral heat content that is defined as before but consideringu⁢(x,t)${u(x,t)}$ a solution to the equation ut=J∗u-u${u_{t}=J\ast u-u}$ inside D with u=0${u=0}$ in ℝN∖D${\mathbb{R}^{N}\setminus D}$ andinitial condition u0=χD${u_{0}=\chi_{D}}$.

Given the contact forms f and g , and the 1-form h , we discuss the existence of a vector field u verifyingℒu⁢(f)=d⁢(u⁢⌟⁢f)+u⁢⌟⁢d⁢f=h.${\mathcal{L}_{u}(f\/)=d(u\,\lrcorner\,f\/)+u\,\lrcorner\,df=h.}$This is closely related to the pullback equation ,where we seek for a diffeomorphism φ satisfyingφ∗⁢(f)=g.${\varphi^{\ast}(f\/)=g.}$

We consider Aharonov–Bohm operators with two poles and prove sharp asymptotics for simple eigenvalues as the poles collapse at an interior point out of nodal lines of the limit eigenfunction.

We consider nonnegative solutions to -Δ⁢u=f⁢(u)${-\Delta u=f(u)}$ in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating and sliding line technique, we prove symmetry and monotonicity properties of the solutions, under very general assumptions on the nonlinearity f . In fact, we provide a unified approach that works in all cases: f⁢(0)<0${f(0)<0}$, f⁢(0)=0${f(0)=0}$ or f⁢(0)>0${f(0)>0}$.Furthermore, we make the effort to deal with nonlinearities f that may be not locally-Lipschitz continuous.We also provide explicit examples showing the sharpness of our assumptions on the nonlinear function f .

We study existence and summability of solutions for elliptic problems with a power-like lower order term and a Hardy potential.We prove that, due to the presence of the lower order term, solutions exist and are more summable under weaker assumptions than those needed for the existence without it.

We present a series of recent results on some new classes of free boundary problems.Differently from the classical literature, the problems considered have either a “nonlocal” feature (e.g., the interaction or/and the interfacial energy may depend on global quantities) or a “nonlinear” flavor (namely, the total energy is the nonlinear superposition of energy components, thus losing the standard additivity and scale invariances of the problem). The complete proofs and the full details of the results presented here are given in [17, 31, 35, 26, 28, 39].

In the present paper we study the Dirichlet problem for an equation involving the 1-Laplacian and a total variation term as reaction.We prove a strong multiplicity result. Namely, we show that for any positive Radon measure concentrated in a set away from the boundary and singular with respect to a certain capacity, there exists an unbounded solution, and measures supported on disjoint sets generate different solutions.These results can be viewed as the analogue for the 1-Laplacian operator of some known multiplicity results which were first obtained by Ireneo Peral, to whom this article is dedicated, and his collaborators.

This article is a survey on boundedness results for stable solutions to semilinear elliptic problems.For these solutions, we present the currently known L∞${L^{\infty}}$ estimates that hold for all nonlinearities.Such estimates are known to hold up to dimension 4.They are expected to be true also in dimensions 5 to 9, but this is still an open problem which has only beenestablished in the radial case.

We describe the theory we developed in recent times concerning two-phasefree boundary problems governed by elliptic operators with forcing terms.Our results range from existence of viscosity solutions to smoothness ofboth solutions and free boundaries. We also discuss some open questions,possible object of future investigation.

We prove the Wloc2⁢s,p${W_{{\mathrm{loc}}}^{2s,p}}$ local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of ℝN${\mathbb{R}^{N}}$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.