We consider an evolution equation arising in the Peierls–Nabarro model for crystal dislocation. We study the evolution of such a dislocation function and show that, at a macroscopic scale, the ...dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior repulsive potential.
We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group
H
n
, whose prototype is the Dirichlet problem for the
p
-fractional subLaplace equation. These ...problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is
p
=
2
, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent
s
goes to 1.
We extend the global compactness result by Struwe (1984) to any fractional Sobolev spaces Ḣs(Ω), for 0<s<N/2 and Ω⊂RN a bounded domain with smooth boundary. The proof is a simple direct ...consequence of the so-called profile decomposition of Gérard (1998).
We study existence, uniqueness, and other geometric properties of the minimizers of the energy functional
where
denotes the total contribution from Ω in the
H
s
norm of
u
and
W
is a double-well ...potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space
. The results collected here will also be useful for forthcoming papers, where the second and the third author will study the Γ-convergence and the density estimates for level sets of minimizers.
We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order s∈(0,1) and summability growth p∈(1,∞), 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.
We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional
p
...-Laplacian operator on the Heisenberg-Weyl group
H
n
. Among other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates.
We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order (s,p), with summability exponent p in (1,∞) and differentiability order s in (0,1), whose ...prototype is the fractional subLaplacian in the Heisenberg group. We present very recent boundedness and regularity estimates (up to the boundary) for the involved weak solutions, and we introduce the nonlocal counterpart of the Perron Method in the Heisenberg group, by recalling some results on the fractional obstacle problem. Throughout the paper we also list various related open problems.
We obtain an improved Sobolev inequality in
H
˙
s
spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing ...sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in
H
˙
s
obtained in Gérard (ESAIM Control Optim Calc Var 3:213–233,
1998
) using the abstract approach of dislocation spaces developed in Tintarev and Fieseler (Concentration compactness. Functional-analytic grounds and applications. Imperial College Press, London,
2007
). We also analyze directly the local defect of compactness of the Sobolev embedding in terms of measures in the spirit of Lions (Rev Mat Iberoamericana 1:145–201,
1985
, Rev Mat Iberoamericana 1:45–121,
1985
). As a model application, we study the asymptotic limit of a family of subcritical problems, obtaining concentration results for the corresponding optimizers which are well known when
s
is an integer (Rey in Manuscr Math 65:19–37,
1989
, Han in Ann Inst Henri Poincaré Anal Non Linéaire 8:159–174,
1991
, Chou and Geng in Differ Integral Equ 13:921–940,
2000
).
We prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of ...fractional elliptic phases, according to the zero set of a modulating coefficient a=a(⋅,⋅). The model case is driven by the following nonlocal double phase operator,∫|u(x)−u(y)|p−2(u(x)−u(y))|x−y|n+spdy+∫a(x,y)|u(x)−u(y)|q−2(u(x)−u(y))|x−y|n+tqdy, where q≥p and a(⋅,⋅)≧0. Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels. By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are Hölder continuous, whereas similar sharp results for the classical local case do require a to be Hölder continuous. To our knowledge, this is the first (regularity) result for nonlocal double phase problems.
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