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We provide fine asymptotics of solutions of fractional elliptic equations at boundary points where the domain is locally conical; that is, corner type singularities appear. Our method relies on a suitable smoothing of the corner singularity…
The fractional Yamabe problem, proposed by Gonz\'{a}lez-Qing (2013, Anal. PDE) is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the…
We consider a variational problem with boundary singularity and Dirichlet condition. We give a blow-up analysis for sequences of solutions of an equation with exponential nonlinearity. Also, we derive a compactness criterion under some…
We study a class of minimization problems for a nonlocal operator involving an external magnetic potential. The notions are physically justified and consistent with the case of absence of magnetic fields. Existence of solutions is obtained…
The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the…
For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Lai-Sang Young's axioms (which imply exponential decay of correlation and the CLT) a local CLT is proven. In fact, a unified version of the local CLT is…
We prove existence of partitions of an open set $\Omega$ with a given number of phases, which minimize the sum of the fractional perimeters of all the phases, with Dirichlet boundary conditions. In two dimensions we show that, if the…
We investigate the vanishing elasticity limit for minimizers of the Landau-de Gennes model with finite energy. By adopting a refined blow-up and covering analysis, we establish the optimal $ L^p $ ($ 1<p<+\infty $) convergence of minimizers…
We investigate a homogenization problem related to a non-local interface energy with a periodic forcing term. We show the existence of planelike minimizers for such energy. Moreover, we prove that, under suitable assumptions on the…
We show that the solution of the Cauchy problem for the classical ode $m \mathbf y''=\mathbf f$ can be obtained as limit of minimizers of exponentially weighted convex variational integrals. This complements the known results about weighted…
We prove the existence of dark monopole solutions in a recently formulated Yang--Mills--Higgs theory model with technical features similar to the classical monopole problems. The solutions are obtained as energy-minimizing static…
We establish the existence and symmetry of all minimizers of a constrained variational problem involving the fractional gradient. This problem is closely connected to some fractional kinetic equations.
We first take into account variational problems with periodic boundary conditions, and briefly recall some sufficient conditions for a periodic solution of the Euler-Lagrange equation to be either a directional, a weak, or a strong local…
In this paper we study the semilinear partial differential equations in the plane the linear part of which is written in a divergence form. The main result is given as a factorization theorem. This theorem states that every weak solution of…
We study existence and uniqueness of bounded solutions to a fractional sublinear elliptic equation with a variable coefficient, in the whole space. Existence is investigated in connection to a certain fractional linear equation, whereas the…
We prove local boundedness of generalized solutions to a large class of variational problems of linear growth including boundary value problems of minimal surface type and models from image analysis related to the procedure of…
We consider a fractional Plateau's problem dealing with sets with prescribed non-local mean curvature. This problem can be seen as a non-local counterpart of the classical Massari's Problem. We obtain existence and regularity results,…
In this paper we introduce uniformly local weak Zygmund type spaces, and obtain an optimal sufficient condition for the existence of solutions to the critical fractional semilinear heat equation.
The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage…
In this paper, we consider a linear fractional differential equation with fractional boundary conditions. First, by obtaining Green's function, we derive the Lyapunov-type inequalities for such boundary value problems. Furthermore, we use…