Local minimizers of nonhomogeneous quasiconvex variational integrals with standard $ p $-growth of the type
<disp-formula> <tex-math id="FE1"> \begin{document}$ w\mapsto \int \leftF(Dw)-f\cdot ...w\right{\,{{\rm{d}}}x} $\end{document} </tex-math></disp-formula>
feature almost everywhere $ \mbox{BMO} $-regular gradient provided that $ f $ belongs to the borderline Marcinkiewicz space $ L(n, \infty) $.
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 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 study the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional subLaplacian in the Heisenberg group. We prove both the existence and ...uniqueness of the solution, and that solutions inherit regularity properties of the obstacle such as boundedness, continuity and Hölder continuity up to the boundary. We also prove some independent properties of weak supersolutions to the class of problems we are dealing with. Armed with the aforementioned results, we finally investigate the Perron–Wiener–Brelot generalized solution by proving its existence for very general boundary data.
Abstract
We establish sharp global regularity results for solutions to nonhomogeneous, nonuniformly elliptic systems with zero boundary conditions imposed only on some part of the boundary of convex ...domains. In particular, we obtain everywhere Lipschitz continuity under borderline Lorentz assumptions on the forcing term, thus positively settling the optimality issue raised in 11.
Local minimizers of nonhomogeneous quasiconvex variational integrals with standard \(p\)-growth of the type $$ w \mapsto \int \leftF(Dw)-f\cdot w\rightdx $$ feature almost everywhere ...\(\mbox{BMO}\)-regular gradient provided that \(f\) belongs to the borderline Marcinkiewicz space \(L(n,\infty)\).
We study the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional subLaplacian in the Heisenberg group. We prove both the existence and ...uniqueness of the solution, and that solutions inherit regularity properties of the obstacle such as boundedness, continuity and H\"older continuity up to the boundary. We also prove some independent properties of weak supersolutions to the class of problems we are dealing with. Armed with the aforementioned results, we finally investigate the Perron-Wiener-Brelot generalized solution by proving its existence for very general boundary data.
We investigate some effects of the lack of compactness in the critical Sobolev embedding by proving that a famous conjecture of Brezis and Peletier \cite{BP89} does still hold in the Heisenberg ...framework: optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at exactly one point which is a critical point of the Robin function (i. e., the diagonal of the regular part of the Green function associated to the involved domain), in clear accordance with the underlying sub-Riemannian geometry. Consequently, a new suitable definition of domains geometrical regular near their characteristic set is introduced. In order to achieve the aforementioned result, we need to combine proper estimates and tools to attack the related CR Yamabe equation with novel feasible ingredients in PDEs and Calculus of Variations which also aim to constitute general independent results in the Heisenberg framework, as for instance a fine asymptotic control of the optimal functions via the Jerison and Lee extremals realizing the equality in the critical Sobolev inequality \cite{JL88}.
We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order \((s,p)\), with summability exponent \(p \in (1,\infty)\) and differentiability exponent ...\(s\in (0,1)\), 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.
