Interior Calder\'on-Zygmund estimates for solutions to general parabolic equations of pp-Laplacian type

We study general parabolic equations of the form $u_t = div A(x,t, u,D u) + div(|F|^{p-2} F)+ f$ whose principal part depends on the solution itself. The vector field $A$ is assumed to have small mean oscillation in $x$, measurable in $t$, Lipschitz continuous in $u$, and its growth in $Du$ is like the $p$-Laplace operator. We establish interior Calder\'on-Zygmund estimates for locally bounded weak solutions to the equations when $p>2n/(n+2)$. This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucial use of the intrinsic geometry method by DiBenedetto \cite{D2} and the maximal function free approach by Acerbi and Mingione \cite{AM}

Boundary regularity for quasilinear elliptic equations with general Dirichlet boundary data

We study global regularity for solutions of quasilinear elliptic equations of the form \div \A(x,u,\nabla u) = \div \F in rough domains $\Omega$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field \A is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $\Omega$, a small BMO condition in $x$ for \A, and an optimal condition for the Dirichlet boundary data.Comment: arXiv admin note: text overlap with arXiv:1810.1249

Interior gradient estimates for quasilinear elliptic equations

We study quasilinear elliptic equations of the form $\text{div} \mathbf{A}(x,u,\nabla u) = \text{div}\mathbf{F}$ in bounded domains in $\mathbb{R}^n$, $n\geq 1$. The vector field $\mathbf{A}$ is allowed to be discontinuous in $x$, Lipschitz continuous in $u$ and its growth in the gradient variable is like the $p$-Laplace operator with $1<p<\infty$. We establish interior $W^{1,q}$-estimates for locally bounded weak solutions to the equations for every $q>p$, and we show that similar results also hold true in the setting of {\it Orlicz} spaces. Our regularity estimates extend results which are only known for the case $\mathbf{A}$ is independent of $u$ and they complement the well-known interior $C^{1,\alpha}$- estimates obtained by DiBenedetto \cite{D} and Tolksdorf \cite{T} for general quasilinear elliptic equations

Interior second derivative estimates for solutions to the linearized Monge--Amp\`ere equation

Let $\Omega\subset \R^n$ be a bounded convex domain and $\phi\in C(\bar\Omega)$ be a convex function such that $\phi$ is sufficiently smooth on $\partial\Omega$ and the Monge--Amp\`ere measure $\det D^2\phi$ is bounded away from zero and infinity in $\Omega$. The corresponding linearized Monge--Amp\`ere equation is \trace(\Phi D^2 u) =f, where $\Phi := \det D^2 \phi ~ (D^2\phi)^{-1}$ is the matrix of cofactors of $D^2\phi$. We prove a conjecture in \cite{GT} about the relationship between $L^p$ estimates for $D^2 u$ and the closeness between $\det D^2\phi$ and one. As a consequence, we obtain interior $W^{2,p}$ estimates for solutions to such equation whenever the measure $\det D^2\phi$ is given by a continuous density and the function $f$ belongs to $L^q(\Omega)$ for some $q> \max{\{p,n\}}$

Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

We study regularity for solutions of quasilinear elliptic equations of the form \div \A(x,u,\nabla u) = \div \F in bounded domains in $\R^n$. The vector field \A is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under a small BMO condition in $x$ for \A. As a consequence, we obtain that $\nabla u$ is in the classical Morrey space \calM^{q,\lambda} or weighted space $L^q_w$ whenever |\F|^{\frac{1}{p-1}} is respectively in \calM^{q,\lambda} or $L^q_w$, where $q$ is any number greater than $p$ and $w$ is any weight in the Muckenhoupt class $A_{\frac{q}{p}}$. In addition, our two-weight estimate allows the possibility to acquire the regularity for $\nabla u$ in a weighted Morrey space that is different from the functional space that the data |\F|^{\frac{1}{p-1}} belongs to

Global W1,pW^{1,p} estimates for solutions to the linearized Monge--Amp\`ere equations

In this paper, we investigate regularity for solutions to the linearized Monge-Amp\`ere equations when the nonhomogeneous term has low integrability. We establish global $W^{1,p}$ estimates for all $p<\frac{nq}{n-q}$ for solutions to the equations with right hand side in $L^q$ where $n/2<q\leq n$. These estimates hold under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant analogues of the global $W^{1,p}$ estimates of N. Winter for fully nonlinear, uniformly elliptic equations

Global optimization using L\'evy flights

This paper studies a class of enhanced diffusion processes in which random walkers perform L\'evy flights and apply it for global optimization. L\'evy flights offer controlled balance between exploitation and exploration. We develop four optimization algorithms based on such properties. We compare new algorithms with the well-known Simulated Annealing on hard test functions and the results are very promising.Comment: 12 pages, 6 figures, 4 algorithms,Proceedings of Second National Symposium on Research, Development and Application of Information and Communication Technology (ICT.rda'04), Hanoi, Sept 24-25, 200

Local gradient estimates for degenerate elliptic equations

This paper is focused on the local interior $W^{1,\infty}$-regularity for weak solutions of degenerate elliptic equations of the form $\text{div}[\mathbf{a}(x,u, \nabla u)] +b(x, u, \nabla u) =0$, which include those of $p$-Laplacian type. We derive an explicit estimate of the local $L^\infty$-norm for the solution's gradient in terms of its local $L^p$-norm. Specifically, we prove \begin{equation*} \|\nabla u\|_{L^\infty(B_{\frac{R}{2}}(x_0))}^p \leq \frac{C}{|B_R(x_0)|}\int_{B_R(x_0)}|\nabla u(x)|^p dx. \end{equation*} This estimate paves the way for our forthcoming work in establishing $W^{1,q}$-estimates (for $q>p$) for weak solutions to a much larger class of quasilinear elliptic equations

Finding Algebraic Structure of Care in Time: A Deep Learning Approach

Understanding the latent processes from Electronic Medical Records could be a game changer in modern healthcare. However, the processes are complex due to the interaction between at least three dynamic components: the illness, the care and the recording practice. Existing methods are inadequate in capturing the dynamic structure of care. We propose an end-to-end model that reads medical record and predicts future risk. The model adopts the algebraic view in that discrete medical objects are embedded into continuous vectors lying in the same space. The bag of disease and comorbidities recorded at each hospital visit are modeled as function of sets. The same holds for the bag of treatments. The interaction between diseases and treatments at a visit is modeled as the residual of the diseases minus the treatments. Finally, the health trajectory, which is a sequence of visits, is modeled using a recurrent neural network. We report preliminary results on chronic diseases - diabetes and mental health - for predicting unplanned readmission.Comment: Accepted NIPS ML4H workshop 201

Resset: A Recurrent Model for Sequence of Sets with Applications to Electronic Medical Records

Modern healthcare is ripe for disruption by AI. A game changer would be automatic understanding the latent processes from electronic medical records, which are being collected for billions of people worldwide. However, these healthcare processes are complicated by the interaction between at least three dynamic components: the illness which involves multiple diseases, the care which involves multiple treatments, and the recording practice which is biased and erroneous. Existing methods are inadequate in capturing the dynamic structure of care. We propose Resset, an end-to-end recurrent model that reads medical record and predicts future risk. The model adopts the algebraic view in that discrete medical objects are embedded into continuous vectors lying in the same space. We formulate the problem as modeling sequences of sets, a novel setting that have rarely, if not, been addressed. Within Resset, the bag of diseases recorded at each clinic visit is modeled as function of sets. The same hold for the bag of treatments. The interaction between the disease bag and the treatment bag at a visit is modeled in several, one of which as residual of diseases minus the treatments. Finally, the health trajectory, which is a sequence of visits, is modeled using a recurrent neural network. We report results on over a hundred thousand hospital visits by patients suffered from two costly chronic diseases -- diabetes and mental health. Resset shows promises in multiple predictive tasks such as readmission prediction, treatments recommendation and diseases progression