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Karthik Adimurthi

Publications and source records attributed to Karthik Adimurthi.

29 records · Page 2Linked to original sources
arXiv2018

Borderline regularity for fully nonlinear equations in Dini domains

In this paper, we prove borderline gradient continuity of viscosity solutions to Fully nonlinear elliptic equations at the boundary of a $C^{1,\dini}$-domain. Our main result Theorem 3.1 is a sharpening of the boundary gradient estimate proved by Ma-Wang following the borderline interior gradient regularity estimates established Daskalopoulos-Kuusi-Mingione. We however mention that, differently from the approach in the interior case which depends on $W^{1,q}$ estimates, our proof is slightly more geometric and is based on compactness arguments inspired by the techniques in the fundamental works of Caffarelli.

math.AP↗
arXiv2018

End point gradient estimates for quasilinear parabolic equations with variable exponent growth on nonsmooth domains

In this paper, we study quasilinear parabolic equations with the nonlinearity structure modeled after the $p(x,t)$-Laplacian on nonsmooth domains. The main goal is to obtain end point Calderón-Zygmund type estimates in the variable exponent setting. In a recent work \cite{byun2016nonlinear}, the estimates obtained were strictly above the natural exponent $p(x,t)$ and hence there was a gap between the natural energy estimates and the estimates above $p(x,t)$ (see \eqref{energy} and \eqref{byunok}). Here, we bridge this gap to obtain the end point case of the estimates obtained in \cite{byun2016nonlinear}. To this end, we make use of the parabolic Lipschitz truncation developed in \cite{KL} and obtain significantly improved a priori estimates below the natural exponent with stability of the constants. An important feature of the techniques used here is that we make use of the unified intrinsic scaling introduced in \cite{adimurthi2018sharp}, which enables us to handle both the singular and degenerate cases simultaneously.

math.AP↗
arXiv2018

Gradient weighted norm inequalities for linear elliptic equations with discontinuous coefficients

Local and global weighted norm estimates involving Muckenhoupt weights are obtained for gradient of solutions to linear elliptic Dirichlet boundary value problems in divergence form over a Lipschitz domain $Ω$. The gradient estimates are obtained in weighted Lebesgue and Lorentz spaces, which also yield estimates in Lorentz-Morrey spaces as well as Hölder continuity of solutions. The significance of the work lies on its applicability to very weak solutions (that belong to $W^{1,p}_{0}(Ω)$ for some $p>1$ but not necessarily in $W^{1,2}_{0}(Ω)$) to inhomogeneous equations with coefficients that may have discontinuities but have a small mean oscillation. The domain is assumed to have a Lipschitz boundary with small Lipschitz constant and as such allows corners. The approach implemented makes use of localized sharp maximal function estimates as well as known regularity estimates for very weak solutions to the associated homogeneous equations. The estimates are optimal in the sense that they coincide with classical weighted gradient estimates in the event the coefficients are continuous and the domain has smooth boundary.

math.AP↗
arXiv2018

Nonlinear equations with gradient natural growth and distributional data, with applications to a Schrödinger type equation

We obtain necessary and sufficient conditions with sharp constants on the distribution $σ$ for the existence of a globally finite energy solution to the quasilinear equation with a gradient source term of natural growth of the form $-Δ_p u = |\nabla u|^p + σ$ in a bounded open set $Ω\subset \mathbb{R}^n$. Here $Δ_p$, $p>1$, is the standard $p$-Laplacian operator defined by $Δ_p u={\rm div}\, (|\nabla u|^{p-2}\nabla u)$. The class of solutions that we are interested in consists of functions $u\in W^{1,p}_0(Ω)$ such that $e^{μ u}\in W^{1,p}_0(Ω)$ for some $μ>0$ and the inequality \begin{equation*} \int_Ω |φ|^p |\nabla u|^p dx \leq A \int_Ω|\nabla φ|^p dx \end{equation*} holds for all $φ\in C_c^\infty(Ω)$ with some constant $A>0$. This is a natural class of solutions at least when the distribution $σ$ is nonnegative. The study of $-Δ_p u = |\nabla u|^p + σ$ is applied to show the existence of globally finite energy solutions to the quasilinear equation of Schrödinger type $-Δ_p v = σ\, v^{p-1}$, $v\geq 0$ in $Ω$, and $v=1$ on $\partialΩ$, via the exponential transformation $u\mapsto v=e^{\frac{u}{p-1}}$.

math.AP↗
arXiv2018

Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers

We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form $-Δ_p u = |\nabla u|^p + σ$ in a bounded domain $Ω\subset \mathbb{R}^n$. Here $Δ_p$, $p>1$, is the standard $p$-Laplacian operator defined by $Δ_p u={\rm div}\, (|\nabla u|^{p-2}\nabla u)$, and the datum $σ$ is a signed distribution in $Ω$. The class of solutions that we are interested in consists of functions $u\in W^{1,p}_0(Ω)$ such that $|\nabla u|\in M(W^{1,p}(Ω)\rightarrow L^p(Ω))$, a space pointwise Sobolev multipliers consisting of functions $f\in L^{p}(Ω)$ such that \begin{equation*} \int_Ω |f|^{p} |φ|^p dx \leq C \int_Ω (|\nabla φ|^p + |φ|^p) dx \quad \forall φ\in C^\infty(Ω), \end{equation*} for some $C>0$. This is a natural class of solutions at least when the distribution $σ$ is nonnegative and compactly supported in $Ω$. We show essentially that, with only a gap in the smallness constants, the above equation has a solution in this class if and only if one can write $σ={\rm div}\, F$ for a vector field $F$ such that $|F|^{\frac{1}{p-1}}\in M(W^{1,p}(Ω)\rightarrow L^p(Ω))$. As an important application, via the exponential transformation $u\mapsto v=e^{\frac{u}{p-1}}$, we obtain an existence result for the quasilinear equation of Schrödinger type $-Δ_p v = σ\, v^{p-1}$, $v\geq 0$ in $Ω$, and $v=1$ on $\partialΩ$, which is interesting in its own right.

math.AP↗
arXiv2018

Gradient weighted estimates at the natural exponent for Quasilinear Parabolic equations

In this paper, we obtain weighted norm inequalities for the spatial gradients of weak solutions to quasilinear parabolic equations with weights in the Muckenhoupt class $A_{\frac{q}{p}}(\mathbb{R}^{n+1})$ for $q\geq p$ on non-smooth domains. Here the quasilinear nonlinearity is modelled after the standard $p$-Laplacian operator. Until now, all the weighted estimates for the gradient were obtained only for exponents $q>p$. The results for exponents $q>p$ used the full complicated machinery of the Calderón-Zygmund theory developed over the past few decades, but the constants blow up as $q \rightarrow p$ (essentially because the Maximal function is not bounded on $L^1$). In order to prove the weighted estimates for the gradient at the natural exponent, i.e., $q=p$, we need to obtain improved a priori estimates below the natural exponent. To this end, we develop the technique of Lipschitz truncation based on \cite{AdiByun2,KL} and obtain significantly improved estimates below the natural exponent. Along the way, we also obtain improved, unweighted Calderón-Zygmund type estimates below the natural exponent which is new even for the linear equations.

math.AP↗
arXiv2018

Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents

We prove boundary higher integrability for the (spatial) gradient of \emph{very weak} solutions of quasilinear parabolic equations of the form $$ \left\{ \begin{array}{ll} u_t - div \mathcal{A}(x,t,\nabla u) = 0 &\quad \text{on} \ Ω\times (-T,T), \\ u = 0 &\quad \text{on} \ \partial Ω\times (-T,T), \end{array} \right. $$ where the non-linear structure $\mathcal{A}(x, t,\nabla u)$ is modelled after the variable exponent $p(x,t)$-Laplace operator given by $|\nabla u|^{p(x,t)-2} \nabla u$. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary by constructing a suitable test function which is Lipschitz continuous and preserves the boundary values. In the interior case, such a result was proved in \cite{bogelein2014very} provided $p(x,t) \geq \mathfrak{p}^- \geq 2$ holds and was then extended to the singular case $\frac{2n}{n+2}< \mathfrak{p}^-\leq p(x,t)\leq \mathfrak{p}^+ \leq 2$ in \cite{li2017very}. This restriction was necessary because the intrinsic scalings for quasilinear parabolic problems are different in the case $\mathfrak{p}^+ \leq 2$ and $\mathfrak{p}^-\geq 2$. In this paper, we develop a new unified intrinsic scaling, using which, we are able to extend the results of \cite{bogelein2014very,li2017very} to the full range $\frac{2n}{n+2} < \mathfrak{p}^- \leq p(x,t)\leq \mathfrak{p}^+<\infty$ and also obtain analogous results upto the boundary. \emph{The main novelty of this paper is that our methods are able to handle both the singular case and degenerate case simultaneously.} To simplify the exposition, we will only prove the higher integrability result near the boundary, provided the domain $Ω$ satisfies a uniform measure density condition. Our techniques are also applicable to higher order equations as well as systems.

math.AP↗
arXiv2018

Boundary higher integrability for very weak solutions of quasilinear parabolic equations

We prove boundary higher integrability for the (spatial) gradient of \emph{very weak} solutions of quasilinear parabolic equations of the form $$u_t - \text{div}\,\mathcal{A}(x,t, \nabla u)=0 \quad \text{on} \ Ω\times \mathbb{R},$$ where the non-linear structure $\text{div}\,\mathcal{A}(x, t,\nabla u)$ is modelled after the $p$-Laplace operator. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary. In order to do this, we construct a suitable test function which is Lipschitz continuous and preserves the boundary values. \emph{These results are new even for linear parabolic equations on domains with smooth boundary and make no assumptions on the smoothness of $\mathcal{A}(x,t,\nabla u)$}. These results are also applicable for systems as well as higher order parabolic equations.

math.AP↗
arXiv2017

Sharp gradient estimates for quasilinear elliptic equations with $p(x)$ growth on nonsmooth domains

In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the $p(x)$-Laplacian on nonsmooth domains and obtain sharp Calderón-Zygmund type estimates in the variable exponent setting. In a recent work of \cite{BO}, the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above $p(x)$, see \eqref{energy_introduction} and \eqref{byun_ok_estimate}. Here, we bridge this gap to obtain the end point case of the estimates obtained in \cite{BO}, see \eqref{our_estimate}. In order to do this, we have to obtain significantly improved a priori estimates below $p(x)$, which is the main contribution of this paper. We also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature.

math.AP↗
arXiv2014

Global Lorentz and Lorentz-Morrey estimates below the natural exponent for quasilinear equations

Lorentz and Lorentz-Morrey estimates are obtained for gradients of very weak solutions to quasilinear equations of the form $$\text{div}\,\mathcal{A}(x, \nabla u)=\text{div}\, |{\bf f}|^{p-2}{\bf f},$$ where $\text{div}\,\mathcal{A}(x, \nabla u)$ is modelled after the $p$-Laplacian, $p>1$. The estimates are global over bounded domains that satisfy a mild exterior uniform thickness condition that involves the $p$-capacity. The vector field datum ${\bf f}$ is allowed to have low degrees of integrability and thus solutions may not have finite $L^p$ energy. A higher integrability result at the boundary of the ground domain is also obtained for infinite energy solutions to the associated homogeneous equations.

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