arXiv subjects
Publications and source records attributed to Riddhi Mishra.
We prove that the boundary of every bounded $(M^{1,p},M^{1,q})$-extension domain in a $Q$-doubling metric measure space has measure zero whenever $0<q<p<\infty$, with the additional restriction that $p<qQ/(Q-q)$ if $q<Q$.
We prove Sobolev interpolation inequalities on extension domains that have a form reminiscent of the corresponding whole-space inequalities. This form is crucial in certain applications, which we discuss as well. The technical key ingredient is the notion of a Lebesgue $W^{1,p}$-extension domain, which we introduce here, and our proof that, for $1<p<\infty$, any $W^{1,p}$-extension domain is a Lebesgue $W^{1,p}$-extension domain.
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $1 < p < \infty$. We prove that there is a bounded extension operator $\dot{W}^{1,p}(\Omega)\to \dot{W}^{1,p}(\mathbb{R}^n)$ if and only if $\Omega$ satisfies the measure density condition and a Bourgain-Brezis-Mironescu type inequality (or limiting formula). As a key ingredient, we establish a fractional Poincar\'e-type inequality under the assumption of Ahlfors regularity alone, improving a result of Ponce (2004). We also prove that, under a mild Hausdorff measure condition on the boundary $\partial \Omega$, fractional extension (from $\dot{W}^{1,p}(\Omega)$ to $\dot{W}^{s,p}(\mathbb{R}^n)$) at a single exponent $s > 1/p$ self-improves to full first-order Sobolev extension (from $\dot{W}^{1,p}(\Omega)$ to $\dot{W}^{1,p}(\mathbb{R}^n)$). These results clarify the role of nonlocal estimates in the geometry of Sobolev extension domains.
We give sufficient geometric conditions, not involving capacities, for a compact null set to be removable for the Sobolev functions on weighted $\mathbb R^n$, defined as the closure of smooth functions in the weighted Sobolev norm. Our porosity conditions are in terms of suitable coverings by cubes. The weights are assumed to be doubling and satisfy a Poincaré inequality, which includes, but is not equal to, the famous class of Muckenhoupt weights. Our proofs use ideas and techniques from the theory of analysis on metric spaces.
We study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, for a certain range of exponents $p$ and $q$, we construct a $(W^{1, p}, W^{1, q})$-extension domain which is not an $(L^{1, p}, L^{1, q})$-extension domain.
In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results. 1- Let $1\leq q\leq p\leq \infty$. Then a bounded $(L^{1, p}, L^{1, q})$-extension domain is also a $(W^{1, p}, W^{1, q})$-extension domain. 2- Let $1\leq q\leq p<q^\star\leq \infty$ or $n< q \leq p\leq \infty$. Then a bounded domain is a $(W^{1, p}, W^{1, q})$-extension domain if and only if it is an $(L^{1, p}, L^{1, q})$-extension domain. 3- For $1\leq q<n$ and $q^\star<p\leq \infty$, there exists a bounded domain $Ω\subset\mathbb{R}^n$ which is a $(W^{1, p}, W^{1, q})$-extension domain but not an $(L^{1, p}, L^{1, q})$-extension domain for $1 \leq q <p\leq n$.
We show that the volume of the boundary of a bounded Sobolev $(p,q)$-extension domain is zero when $1\leq q <p< \frac{qn}{(n-q)}.$
In this paper, we show that the existence of a positive weak solution to the equation $(-Δ_g)^s u=f u^{-q(x)}\;\mbox{in}\; Ω,$ where $Ω$ is a smooth bounded domain in $R^N$, $q\in C^1(\overlineΩ)$, and $(-Δ_g)^s$ is the fractional $g$-Laplacian with $g$ is the antiderivative of a Young function and $f$ in suitable Orlicz space subjected to zero Dirichlet condition. This includes the mixed fractional $(p,q)-$Laplacian as a special case. The solution so obtained is also shown to be locally Hölder continuous.