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Register a new userOn asymptotic expansions for the fractional infinity Laplacian
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We propose two asymptotic expansions of the two interrelated integral-type averages, in the context of the fractional $infty$-Laplacian $Delta_infty^s$ for $sin (frac{1}{2},1)$. This operator has been introduced and first studied in [Bjorland-Caffarelli-Figalli, 2012]. Our expansions are parametrised by the radius of the removed singularity $epsilon$, and allow for the identification of $Delta_infty^sphi(x)$ as the $epsilon^{2s}$-order coefficient of the deviation of the $epsilon$-average from the value $phi(x)$, in the limit $epsilonto 0+$. The averages are well posed for functions $phi$ that are only Borel regular and bounded.
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In this paper we obtain the precise description of the asymptotic behavior of the solution $u$ of $$ partial_t u+(-Delta)^{frac{theta}{2}}u=0quadmbox{in}quad{bf R}^Ntimes(0,infty), qquad u(x,0)=varphi(x)quadmbox{in}quad{bf R}^N, $$ where $0<theta<2$ and $varphiin L_K:=L^1({bf R}^N,,(1+|x|)^K,dx)$ with $Kge 0$. Furthermore, we develop the arguments in [15] and [18] and establish a method to obtain the asymptotic expansions of the solutions to a nonlinear fractional diffusion equation $$ partial_t u+(-Delta)^{frac{theta}{2}}u=|u|^{p-1}uquadmbox{in}quad{bf R}^Ntimes(0,infty), $$ where $0<theta<2$ and $p>1+theta/N$.
Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well known equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are now-days available. The nonlinear case, corresponding to the $p$-Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the emph{nonlocal, nonlinear mean value kernel}, by means of which we obtain an asymptotic representation formula for harmonic functions in the viscosity sense, with respect to the fractional (variational) $p$-Laplacian (for $pgeq 2$) and to other gradient dependent nonlocal operators.
We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, with a logistic type reaction depending on a positive parameter. In the subdiffusive and equidiffusive cases, we prove existence and uniqueness of the positive solution when the parameter lies in convenient intervals. In the superdiffusive case, we establish a bifurcation result. A new strong comparison result, of independent interest, plays a crucial role in the proof of such bifurcation result.
We present a novel definition of variable-order fractional Laplacian on Rn based on a natural generalization of the standard Riesz potential. Our definition holds for values of the fractional parameter spanning the entire open set (0, n/2). We then discuss some properties of the fractional Poissons equation involving this operator and we compute the corresponding Green function, for which we provide some instructive examples for specific problems.
In this paper, we prove several Liouville type results for a nonlinear equation involving infinity Laplacian with gradient of the form $$Delta^gamma_infty u + q(x)cdot abla{u} | abla{u}|^{2-gamma} + f(x, u),=,0quad text{in}; mathbb{R}^d,$$ where $gammain [0, 2]$ and $Delta^gamma_infty$ is a $(3-gamma)$-homogeneous operator associated with the infinity Laplacian. Under the assumptions $liminf_{|x|toinfty}lim_{sto0}f(x,s)/s^{3-gamma}>0$ and $q$ is a continuous function vanishing at infinity, we construct a positive bounded solution to the equation and if $f(x,s)/s^{3-gamma}$ decreasing in $s$, we also obtain the uniqueness. While, if $limsup_{|x|toinfty}sup_{[delta_1,delta_2]}f(x,s)<0$, then nonexistence result holds provided additionally some suitable conditions. To this aim, we develop new technique to overcome the degeneracy of infinity Laplacian and nonlinearity of gradient term. Our approach is based on a new regularity result, the strong maximum principle, and Hopfs lemma for infinity Laplacian involving gradient and potential. We also construct some examples to illustrate our results. We further study the related Dirichlet principal eigenvalue of the corresponding nonlinear operator $$Delta^gamma_infty u + q(x)cdot abla{u} | abla{u}|^{2-gamma} + c(x)u^{3-gamma},$$ in smooth bounded domains, which may be considered as of independent interest. Our results could be seen as the extension of Liouville type results obtained by Savin [48] and Ara{u}jo et. al. [1] and a counterpart of the uniqueness obtained by Lu and Wang [39,40] for sign-changing $f$.
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