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Register a new userEstimates for the $infty$-Laplacian relative to vector fields
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In this paper we prove a Holder regularity estimate for viscosity solutions of inhomogeneous equations governed by the infinite Laplace operator relative to a frame of vector fields.
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In this paper, we apply blow-up analysis and Liouville type theorems to study pointwise a priori estimates for some quasilinear equations with p-Laplace operator. We first obtain pointwise interior estimates for the gradient of p-harmonic function, i.e., the solution of $Delta_{p}u=0, xinOmega$, which extends the well-established results of the interior estimates of the gradient of harmonic function. We then get singularity and decay estimates of the sign changing solution of Lane-Emden-Fowler type p-Laplace equation $-Delta_{p}u=|u|^{lambda-1}u, xinOmega$, which are then generalized for the equation with general right hand term $f(x,u)$, under some asymptotic conditions of $f$. Lastly, we get pointwise estimates for higher order derivatives of the solution of $-Delta u=u^{lambda},xinOmega$, the case of $p=2$ for p-Laplace equation.
In this paper we present a new bootstrap procedure for elliptic systems with two unknown functions. Combining with the $L^p$-$L^q$-estimates, it yields the optimal $L^infty$-regularity conditions for the three well-known types of weak solutions: $H_0^1$-solutions, $L^1$-solutions and $L^1_delta$-solutions. Thanks to the linear theory in $L^p_delta(Omega)$, it also yields the optimal conditions for a priori estimates for $L^1_delta$-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.
For weak solutions to the evolutional $p$-Laplace equation with a time-dependent Radon measure on the right hand side we obtain pointwise estimates via a nonlinear parabolic potential.
Multi-armed bandits are widely applied in scenarios like recommender systems, for which the goal is to maximize the click rate. However, more factors should be considered, e.g., user stickiness, user growth rate, user experience assessment, etc. In this paper, we model this situation as a problem of $K$-armed bandit with multiple losses. We define relative loss vector of an arm where the $i$-th entry compares the arm and the optimal arm with respect to the $i$-th loss. We study two goals: (a) finding the arm with the minimum $ell^infty$-norm of relative losses with a given confidence level (which refers to fixed-confidence best-arm identification); (b) minimizing the $ell^infty$-norm of cumulative relative losses (which refers to regret minimization). For goal (a), we derive a problem-dependent sample complexity lower bound and discuss how to achieve matching algorithms. For goal (b), we provide a regret lower bound of $Omega(T^{2/3})$ and provide a matching algorithm.
Consider the Landau equation with Coulomb potential in a periodic box. We develop a new $L^{2}rightarrow L^{infty }$ framework to construct global unique solutions near Maxwellian with small $L^{infty } $norm. The first step is to establish global $L^{2}$ estimates with strong velocity weight and time decay, under the assumption of $L^{infty }$ bound, which is further controlled by such $L^{2}$ estimates via De Giorgis method cite{golse2016harnack} and cite{mouhot2015holder}. The second step is to employ estimates in $S_{p}$ spaces to control velocity derivatives to ensure uniqueness, which is based on Holder estimates via De Giorgis method cite{golse2016harnack}, cite{golse2015holder}, and cite{mouhot2015holder}.
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