Potential Analysis最新文献

We provide sharp two-sided estimates of the heat kernel of the Dirichlet fractional Laplacian on the half-line perturbed by a Hardy potential.

In this paper, we investigate a class of doubly nonlinear evolutions PDEs. We establish sharp regularity for the solutions in Hölder spaces. The proof is based on the geometric tangential method and intrinsic scaling technique. Our findings extend and recover the results in the context of the classical evolution PDEs with singular signature via a unified treatment in the slow, normal and fast diffusion regimes. In addition, we provide some applications to certain nonlinear evolution models, which may have their own mathematical interest.

Let (u_{s}) denote a solution of the fractional Poisson problem

where (Nge 2) and (Omega subset {mathbb {R}}^{N}) is a bounded domain of class (C^{2}). We show that the solution mapping (smapsto u_{s}) is differentiable in (L^infty (Omega )) at s = 1, namely, at the nonlocal-to-local transition. Moreover, using the logarithmic Laplacian, we characterize the derivative (partial _{s} u_{s}) as the solution to a boundary value problem. This complements the previously known differentiability results for s in the open interval (0, 1). Our proofs are based on an asymptotic analysis to describe the collapse of the nonlocality of the fractional Laplacian as s approaches 1. We also provide a new representation of (partial _{s} u_{s}) for s (in (0,1)) which allows us to refine previously obtained Green function estimates.

We consider the asymptotics of the discrete heat kernel on isoradial graphs for the case where the time and the edge lengths tend to zero simultaneously. Depending on the asymptotic ratio between time and edge lengths, we show that two different regimes arise: (i) a Gaussian regime and (ii) a Poissonian regime, which resemble the short-time asymptotics of the heat kernel on (i) Euclidean spaces and (ii) graphs, respectively.

We consider the spaces ({text {L}}^p(X,nu ;V)), where X is a separable Banach space, (mu ) is a centred non-degenerate Gaussian measure, (nu :=Ke^{-U}mu ) with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions (Fin W^{1,p}(X,nu ;V)), which allows us to show that for every (pin (1,infty )) and every (kin mathbb {N}) the norm in (W^{k,p}(X,nu )) is equivalent to the graph norm of (D_H^{k}) (the k-th Malliavin derivative) in ({text {L}}^p(X,nu )). To conclude, we show exponential decay estimates for the V-valued perturbed Ornstein-Uhlenbeck semigroup ((T^V(t))_{tge 0}), defined in Section 2.6, as t goes to infinity. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck ((T(t))_{tge 0}), and pointwise estimates for (|D_HT(t)f|_H^p) by means of both (T(t)|D_Hf|^p_H) and (T(t)|f|^p).

Let ((X, d, mu )) be a metric space with a metric d and a doubling measure (mu ). Assume that the operator L generates a bounded holomorphic semigroup (e^{-tL}) on (L^2(X)) whose semigroup kernel satisfies the Gaussian upper bound. Also assume that L has a bounded holomorphic functional calculus on (L^2(X)). Then the Hardy spaces (H^p_L(X)) associated with the operator L can be defined for (0 < p le 1). In this paper, we revisit the Calderón-Zygmund decomposition and show that a function (f in L^1(X)cap L^2(X)) can be decomposed into a good part which is an (L^{infty }) function and a bad part which is in (H^p_L(X)) for some (0< p <1). An important result of our variants of Calderón-Zygmund decompositions is that if a sub-linear operator T is bounded from (L^r(X)) to (L^r(X)) for some (r > 1) and also bounded from (H^p_L(X)) to (L^p(X)) for some (0< p < 1), then T is of weak type (1, 1) and bounded from (L^q(X)) to (L^q(X)) for all (1< q <r).

In this work, we develop weighted Lorentz-Sobolev estimates for viscosity solutions of fully nonlinear elliptic equations with oblique boundary condition under weakened convexity conditions in the following configuration:

where (Omega ) is a bounded domain in (mathbb {R}^{n}) ((nge 2)), under suitable assumptions on the source term f, data (beta , gamma ) and g. In addition, we obtain Lorentz-Sobolev estimates for solutions to the obstacle problem and others applications.

We show that submanifolds with infinite mean exit time can not be isometrically and minimally immersed into cylinders, horocylinders, cones, and wedges of some product spaces. Our approach is not based on the weak maximum principle at infinity, and thus it permits us to generalize previous results concerning non-immersibility of stochastically complete submanifolds. We also produce estimates for the complete tower of moments for submanifolds with small mean curvature immersed into cylinders.

We present a proof of scale-invariant boundary Harnack principle for uniform domains when the underlying space satisfies a scale-invariant elliptic Harnack inequality. Our approach does not assume the underlying space to be geodesic. Additionally, the existence of Green functions is also not assumed beforehand and is ensured by a recent result from M. T. Barlow, Z.-Q. Chen and M. Murugan.

Let ((Sigma ,g)) be a closed Riemann surface, (lambda _1(Sigma )) be the first eigenvalue of the Laplace-Beltrami operator. Assume (h:Sigma rightarrow mathbb {R}) is some smooth sign-changing function. Using blow-up analysis, we prove that for any (alpha <lambda _1(Sigma )), the supremum

is attained by some admissible function (u_alpha ). This generalizes earlier results of Yang (J. Differential Equations 2015) and Hou (J. Math. ineq. 2018). Our result resembles existence of solutions to the mean field equations

where h is a smooth sign-changing function. Such problems were extensively studied by L. Sun and J. Y. Zhu (Cal. Var. 2021; arXiv: 2012.12840).