We deal with the obstacle problem for a class of nonlinear integro-differential operators, whose model is the fractional $p$-Laplacian with measurable coeffcients. In accordance with well-known ...results for the analog for the pure fractional Laplacian operator, the corresponding solutions inherit regularity properties from the obstacle, both in the case of boundedness, continuity, and Hölder continuity, up to the boundary.
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.
We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group \(\mathbb{H}^n\), 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 consider nonlinear parabolic equations of the type
under standard growth conditions on
a
, with
f
only assumed to be integrable. We prove general decay estimates up to the boundary for level sets ...of the solutions
u
and the gradient
Du
which imply very general estimates in Lebesgue and Lorentz spaces. Assuming only that the involved domains satisfy a mild exterior capacity density condition, we provide global regularity results.
We investigate some of the effects of the lack of compactness in the critical Folland-Stein-Sobolev embedding in very general (possible non-smooth) domains, by proving via De Giorgi's ...\(\Gamma\)-convergence techniques that optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point. In the second part of the paper, we try to restore the compactness by extending the celebrated Global Compactness result to the Heisenberg group via a completely different approach with respect to the original one by Struwe 37.
We deal with a wide class of kinetic equations, $$ \big \partial_t + v\cdot\nabla_x\big f = \mathcal{L}_v f. $$ Above, the diffusion term \(\mathcal{L}_v\) is an integro-differential operator, whose ...nonnegative kernel is of fractional order \(s\in(0,1)\) having merely measurable coefficients. Amongst other results, we are able to prove that nonnegative weak solutions \(f\) do satisfy $$ \sup_{Q^-} f \ \leq \ c\inf_{Q^+} f, $$ where \(Q^{\pm}\) are suitable slanted cylinders. No a-priori boundedness is assumed, as usually in the literature, since we are also able to prove a general interpolation inequality in turn giving local boundedness which is valid even for weak subsolutions with no sign assumptions. To our knowledge, this is the very first time that a strong Harnack inequality is proven for kinetic integro-differential-type equations. A new independent result, a Besicovitch-type covering argument for very general kinetic geometries, is also stated and proved.
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 $\mathbb{H}^n$. Amongst other results, we
prove that the weak solutions to such a class of problems are bounded and
H\"older continuous, by also establishing general estimates as fractional
Caccioppoli-type estimates with tail and logarithmic-type estimates.
We extend the Global Compactness result by M. Struwe (Math. Z, 1984) to any fractional Sobolev spaces \(\dot{H}^s(\Omega)\) for \(0<s<N/2\) and \(\Omega \subset \mathbb{R}^N\) a bounded domain with ...smooth boundary. The proof is a simple direct consequence of the so-called Profile Decomposition of P. Gerard (ESAIM: Control, Optimisation and Calculus of Variations, 1998).
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