Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRegularity results of nonlinear perturbed stable-like operators
71
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider a class of fully nonlinear integro-differential operators where the nonlocal integral has two components: the non-degenerate one corresponds to the $alpha$-stable operator and the second one (possibly degenerate) corresponds to a class of textit{lower order} Levy measures. Such operators do not have a global scaling property. We establish H{o}lder regularity, Harnack inequality and boundary Harnack property of solutions of these operators.
rate research
Read More
We study solutions to $Lu=f$ in $Omegasubsetmathbb R^n$, being $L$ the generator of any, possibly non-symmetric, stable Levy process. On the one hand, we study the regularity of solutions to $Lu=f$ in $Omega$, $u=0$ in $Omega^c$, in $C^{1,alpha}$ domains~$Omega$. We show that solutions $u$ satisfy $u/d^gammain C^{varepsilon_circ}big(overlineOmegabig)$, where $d$ is the distance to $partialOmega$, and $gamma=gamma(L, u)$ is an explicit exponent that depends on the Fourier symbol of operator $L$ and on the unit normal $ u$ to the boundary $partialOmega$. On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the non-symmetric setting presents some new interesting features. Finally, we generalize the integration by parts identities in half spaces to the case of bounded $C^{1,alpha}$ domains. We do it via a new efficient approximation argument, which exploits the Holder regularity of $u/d^gamma$. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.
In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form $displaystyle -operatorname{div}(A(| abla u|) abla u)+Bleft( | abla u|right) =f(u)$; in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin.
In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre cite{CS1} are extended to those for the integro-differential operators associated with symmetric, regularly varying kernels at zero. In particular, we obtain the uniform Harnack inequality and Holder estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels $K_{sigma, beta}$ satisfying $$ K_{sigma,beta}(y)asymp frac{ 2-sigma}{|y|^{n+sigma}}left( logfrac{2}{|y|^2}right)^{beta(2-sigma)}quad mbox{near zero} $$ with respect to $sigmain(0,2)$ close to $2$ (for a given $betainmathbb R$), where the regularity estimates do not blow up as the order $ sigmain(0,2)$ tends to $2.$
We establish Ambrosetti--Prodi type results for viscosity and classical solutions of nonlinear Dirichlet problems for the fractional Laplace and comparable operators. In the choice of nonlinearities we consider semi-linear and super-linear growth cases separately. We develop a new technique using a functional integration-based approach, which is more robust in the non-local context than a purely analytic treatment.
In this paper we characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension $2$. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree $2$. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential operators, which can be completely characterized in terms of the asymptotic behavior of the Weyl symbol. In conclusion we observe that we have obtained a new class of globally regular partial differential operators which is disjoint from the class of hypo-elliptic operators in the sense of Shubin.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
