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Register a new userGlobal gradient estimates for very singular quasilinear elliptic equations with measure data
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This paper continues the development of regularity results for quasilinear measure data problems begin{align*} begin{cases} -mathrm{div}(A(x, abla u)) &= mu quad text{in} Omega, quad quad qquad u &=0 quad text{on} partial Omega, end{cases} end{align*} in Lorentz and Lorentz-Morrey spaces, where $Omega subset mathbb{R}^n$ ($n ge 2$), $mu$ is a finite Radon measure on $Omega$, and $A$ is a monotone Caratheodory vector valued operator acting between $W^{1,p}_0(Omega)$ and its dual $W^{-1,p}(Omega)$. It emphasizes that this paper studies the `very singular case $1<p le frac{3n-2}{2n-1}$ and the problem is considered under the weak assumption, where the $p$-capacity uniform thickness condition is imposed on the complement of domain $Omega$. There are two main results obtained in our study pertaining to the global gradient estimates of solutions in Lorentz and Lorentz-Morrey spaces involving the use of maximal and fractional maximal operators. The idea for writing this working paper comes directly from the recent results by others in the same research topic, where global estimates for gradient of solutions for the `very singular case still remains a challenge, specifically related to Lorentz and Lorentz-Morrey spaces.
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We deal with a global Calderon-Zygmund type estimate for elliptic obstacle problems of $p$-Laplacian type with measure data. For this paper, we focus on the singular case of growth exponent, i.e. $1<p le 2-frac{1}{n}$. In addition, the emphasis of this paper is in obtaining the Lorentz bounds for the gradient of solutions with the use of fractional maximal operators.
This paper is focused on the local interior $W^{1,infty}$-regularity for weak solutions of degenerate elliptic equations of the form $text{div}[mathbf{a}(x,u, abla u)] +b(x, u, abla u) =0$, which include those of $p$-Laplacian type. We derive an explicit estimate of the local $L^infty$-norm for the solutions gradient in terms of its local $L^p$-norm. Specifically, we prove begin{equation*} | abla u|_{L^infty(B_{frac{R}{2}}(x_0))}^p leq frac{C}{|B_R(x_0)|}int_{B_R(x_0)}| abla 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.
In this note we establish existence and uniqueness of weak solutions of linear elliptic equation $text{div}[mathbf{A}(x) abla u] = text{div}{mathbf{F}(x)}$, where the matrix $mathbf{A}$ is just measurable and its skew-symmetric part can be unbounded. Global reverse H{o}lders regularity estimates for gradients of weak solutions are also obtained. Most importantly, we show, by providing an example, that boundedness and ellipticity of $mathbf{A}$ is not sufficient for higher integrability estimates even when the symmetric part of $mathbf{A}$ is the identity matrix. In addition, the example also shows the necessity of the dependence of $alpha$ in the H{o}lder $C^alpha$-regularity theory on the textup{BMO}-semi norm of the skew-symmetric part of $mathbf{A}$. The paper is an extension of classical results obtained by N. G. Meyers (1963) in which the skew-symmetric part of $mathbf{A}$ is assumed to be zero.
We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivation of this paper comes from the study on an abstract result in the viewpoint of the fractional maximal distributions and this work also extends some regularity results proved in cite{PN_dist} by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimates of the gradient of weak solutions via fractional maximal operators and Riesz potential of data. Moreover, in the setting of the paper, we are led to the study of problems with nonlinearity is supposed to be partially weak BMO condition (is measurable in one fixed variable and only satisfies locally small-BMO seminorms in the remaining variables).
Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) ${mathbb L} u = u^p +lambda mu$ in a bounded domain $Omega$ with homogeneous boundary or exterior Dirichlet condition, where $p>1$ and $lambda>0$. The operator ${mathbb L}$ belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum $mu$ is taken in the optimal weighted measure space. The interplay between the operator ${mathbb L}$, the source term $u^p$ and the datum $mu$ yields substantial difficulties and reveals the distinctive feature of the problem. We develop a new unifying technique based on a fine analysis on the Green kernel, which enables us to construct a theory for semilinear equation (E) in measure frameworks. A main thrust of the paper is to provide a fairly complete description of positive solutions to the Dirichlet problem for (E). In particular, we show that there exist a critical exponent $p^*$ and a threshold value $lambda^*$ such that the multiplicity holds for $1<p<p^*$ and $0<lambda<lambda^*$, the uniqueness holds for $1<p<p^*$ and $lambda=lambda^*$, and the nonexistence holds in other cases. Various types of nonlocal operator are discussed to exemplify the wide applicability of our theory.
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