arXiv subjects

Adimurthi

Publications and source records attributed to Adimurthi.

At least 19 recordsLinked to original sources

arXiv2026

Leray-Trudinger Type Exponential Integrability in Log-Weighted Sobolev Spaces

In this article, we conduct a comprehensive study of weighted Sobolev spaces with logarithmic weights, orginially introduced by Calanchi and Ruf to analyze the sharp exponential integrability of radial functions belonging to these spaces. By exploring the connection between these logarithmically weighted energies and the Leray energy, we expand the framework to incorporate non-radial functions. More precisely, we establish optimal exponential integrability for general functions in the spirit of optimal Leray-Trudinger inequalities established by Di Blasio, Pisante and Psaradakis. Furthermore, we prove sharp versions of these inequalities when restricted to radial functions. Notably, the inequalities presented here are fundamentally different in nature from those of Calanchi and Ruf, for which the non-radial extension fails to hold.

math.AP↗

arXiv2024

The Trudinger type inequality in fractional boundary Hardy inequality

We establish Trudinger-type inequality in the context of fractional boundary Hardy-type inequality for the case $sp=d$, where $p>1, ~ s \in (0,1)$ on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$. In particular, we establish fractional version of Trudinger-type inequality with an extra singular function, namely $d$-th power of the distance function from $\partial \Omega$ in the denominator of the integrand. The case $d=1$, as it falls in the category $sp=1$, becomes more delicate where an extra logarithmic correction is required together with subtraction of an average term.

math.AP↗

arXiv2024

Boundary fractional Hardy's inequality in dimension one: The critical case

We prove fractional boundary Hardy's inequality in dimension one for the critical case $sp =1$. Optimality of the inequality is obtained for any $p$. The extra logarithmic correction term appears in usual fashion. We also provide a concrete (workable) example of a sequence of smooth functions that converges to constant function in $W^{s,p}((0,1))$ for $sp=1$ and $p=2$.

math.AP↗

arXiv2024

Fractional Hardy inequality with singularity on submanifold

We establish fractional Hardy inequality on bounded domains in $\mathbb{R}^{d}$ with inverse of distance function from smooth boundary of codimension $k$, where $k=2, \dots,d$, as weight function. The case $sp=k$ is the critical case, where optimal logarithmic corrections are required. All the other cases of $sp k$ are also addressed.

math.AP↗

arXiv2024

Boundary Hardy inequality on functions of bounded variation

Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \infty, \ ~\Omega$ is bounded Lipschitz domain, then for all $u \in C^{\infty}_{c}(\Omega)$, $$\int_{\Omega} \frac{|u(x)|^{p}}{\delta^{p}_{\Omega}(x)} dx \leq C\int_{\Omega} |\nabla u(x) |^{p}dx,$$ where $\delta_\Omega(x)$ is the distance function from $\Omega^c$. In this article, we address the long standing open question on the case $p=1$ by establishing appropriate boundary Hardy inequalities in the space of functions of bounded variation. We first establish appropriate inequalities on fractional Sobolev spaces $W^{s,1}(\Omega)$ and then Brezis, Bourgain and Mironescu's result on limiting behavior of fractional Sobolev spaces as $s\rightarrow 1^{-}$ plays an important role in the proof. Moreover, we also derive an infinite series Hardy inequality for the case $p=1$.

math.AP↗

arXiv2023

Fractional boundary Hardy inequality for the critical cases

We establish generalised fractional boundary Hardy-type inequality, in the spirit of Caffarelli-Kohn-Nirenberg inequality for different values of $s$ and $p$ on various domains in $\mathbb{R}^d, ~ d \geq 1$. In particular, for Lipschitz bounded domains any values of $s$ and $p$ are admissible, settling all the cases in subcritical, supercritical and critical regime. In this paper we have solved the open problems posed by Dyda for the critical case $sp =1$. Moreover we have proved the embeddings of $W^{s,p}_{0}(\Omega)$ in subcritical, critical and supercritical uniformly without using Dyda's decomposition. Additionally, we extend our results to include a weighted fractional boundary Hardy-type inequality for the critical case.

math.AP↗

arXiv2020

Exact and optimal controllability for scalar conservation laws with discontinuous flux

This paper describes the reachable set and resolves an optimal control problem for the scalar conservation laws with discontinuous flux. We give a necessary and sufficient criteria for the reachable set. A new backward resolution has been described to obtain the reachable set. Regarding the optimal control problem we first prove the existence of a minimizer and then the backward algorithm allows us to compute it. The same method also applies to compute the initial data control for an exact control problem. Our methodology for the proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics curves.

math.AP↗

arXiv2018

Single shock solution for non convex scalar conservation laws

In this paper we study the finite time emergence of one shock for the solution of scalar conservation laws in one space dimension with general flux f . We give a necessary and sufficient condition to the initial data connecting to flux. The proof relies on the structure theorem for the linear degenerate flux and the finer analysis of characteristic curves.

math.AP↗

arXiv2017

Positive solutions to a fractional equation with singular nonlinearity

In this paper, we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain $\Omega\subset \R^N$, $N> 2s$: % \begin{eqnarray*} (P_\lambda)\left\{\begin{array}{lll} &(-\Delta)^s u=\lambda(K(x)u^{-\delta}+f(u))\mbox{ in }\Omega &u>0 \mbox{ in }\Omega & u\equiv\, 0\mbox{ in }\R^N\backslash\Omega. \end{array}\right. \end{eqnarray*} % Here $0 0$, $\lambda>0$ and $f\,:\, \R^+\to\R^+$ is a positive $C^2$ function. $K\,:\, \Omega\to \R^+$ is a H\"older continuous function in $\Omega$ which behave as ${\rm dist}(x,\partial\Omega)^{-\beta}$ near the boundary with $0\leq \beta<2s$. First, for any $\delta>0$ and for $\lambda>$ small enough, we prove the existence of solutions to $(P_\lambda)$. Next, for a suitable range of values of $\delta$, we show the existence of an unbounded connected branch of solutions to $(P_\lambda)$ emanating from the trivial solution at $\lambda=0$. For a certain class of nonlinearities $f$, we derive a global multiplicity result that extends results proved in \cite{peral-al}. To establish the results, we prove new properties which are of independent interest and deal with the behavior and H\"older regularity of solutions to $(P_\lambda)$.

math.AP↗

arXiv2016

Exact and optimal controllability for scalar conservation laws with discontinuous flux

This paper deals with an optimal control problem and describes the reachable set for the scalar 1-D conservation laws with discontinuous flux. Regarding the optimal control problem we first prove the existence of a minimizer and then we prescribe an algorithm to compute it. The same method also applies to compute the initial data control. The proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics.

math.AP↗

arXiv2015

Uniqueness of positive solutions of a $n$-Laplace equation in a ball in $\mathbb{r}^n$ with exponential nonlinearity

Let $n \geq 2$ and $\Omega \subset \mathbb{R}^n$ be a bounded domain. Then by Trudinger-Moser embedding, $W_0^{1,n}(\Omega)$ is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semi linear $n$-Laplace equation with critical or sub-critical exponential nonlinearity in a ball $B(R)$ with dirichlet boundary condition. In this paper, we prove that under suitable growth conditions on the nonlinearity, there exists an $\gamma_0 > 0$, and a corresponding $R_0(\gamma_0 ) > 0$ such that for all $0 < R < R_0$ , the problem admits a unique non degenerate positive radial solution $u$ with $\|u\|_{\infty}\geq \gamma_0$.

math.AP↗

arXiv2014

Defect of compactness in spaces of bounded variation

Defect of compactness for non-compact imbeddings of Banach spaces can be expressed in the form of a profile decomposition. This paper extends the profile decomposition for Sobolev spaces proved by Solimini (AIHP 1995) to the non-reflexive case p=1. Since existence of concentration profiles relies on weak-star compactness, the corresponding result is set in a larger, conjugate, space of functions of bounded variation. We prove existence of minimizers for related inequalities and generalizations for to spaces of bounded variation on Lie groups.

math.FA↗

arXiv2014

On the Brezis-Lieb Lemma without pointwise convergence

Brezis-Lieb lemma is a refinement of Fatou lemma providing an evaluation of the gap between the integral for a sequence and the integral for its pointwise limit. This note studies the question if such gap can be evaluated when there is no a.e. convergence. In particular, it gives the same lower bound for the gap in L^p as the gap in the Brezis-Lieb lemma (including the case vector-valued functions) provided that p is greater or equal than 3 and the sequence converges both weakly and weakly in the sense of a duality map. It also shows that the statement is false if p<3. An application is given in form of a Brezis-Lieb lemma for gradients.

math.FA↗

arXiv2013

Second order scheme for scalar conservation law with discontinuous flux

Burger et al.in \cite{karlsen-1} proposed a flux TVD (FTVD) second order scheme by using a new non local limiter algorithm for conservation laws with discontinuous flux modeling clarifier thickener units. In this work we show that their idea of constructing FTVD second order schemes also can be used to construct second order schemes satisfying (A,B)-entropy condition for the scalar conservation law with discontinuous flux with proper modification at the interface. We present numerical experiments to show the superiority of the second order schemes over the monotone first order schemes. We show further from numerical experiments that solutions from these schemes are comparable with the second order schemes obtained from minimod limiter.

math.AP↗

arXiv2011

On compactness in the Trudinger-Moser inequality

The paper studies continutity of Moser nonlinearity in two dimensions with respect to weak convergence. Unlike the critical nonlinearity in the Sobolev inequality, which lacks weak continuity at any point, Moser functional fails to be weakly continuous only in an exceptional case of a concentrating sequence of functions from the Moser family (up to translations and the remainder vanishing in Sobolev norm). The argument is based on a structural description of the defect of weak converegence, analogous to the profile decomposition established by Solimini for Sobolev inequalities, but involving gauge operators specific for the two-dimensional case.

math.AP↗

arXiv2011

Finer analysis of characteristic curves and its application to shock profile, exact and optimal controllability of a scalar conservation law with strict convex flux

Here we consider scalar conservation law in one space dimension with strictly convex flux. Goal of this paper is to study two problems. First problem is to know the profile of the entropy solution. In spite of the fact that, this was studied extensively in last several decades, the complete profile of the entropy solution is not well understood. Second problem is the exact controllability. This was studied for Burgers equation and some partial results are obtained for large time. It was a challenging problem to know the controllability for all time and also for general convex flux. In a seminal paper, Dafermos introduces the characteristic curves and obtain some qualitative properties of a solution of a convex conservation law. In this paper, we further study the finer properties of these characteristic curves. As a bi-product we solve these two problems in complete generality. In view of the explicit formulas of Lax - Oleinik, Joseph - Gowda, target functions must satisfy some necessary conditions. In this paper we prove that it is also sufficient. Method of the proof depends highly on the characteristic methods and explicit formula given by Lax - Oleinik and the proof is constructive. This method allows to solve the optimal controllability problem in a trackable way.

math.AP↗

arXiv2009

On the best constant of Hardy-Sobolev Inequalities

We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant coincides with the best Sobolev constant.

math.AP↗

arXiv2009

On a version of Trudinger-Moser inequality with Möbius shift invariance

The paper raises a question about the optimal critical nonlinearity for the Sobolev space in two dimensions, connected to loss of compactness, and discusses the pertinent concentration compactness framework. We study properties of the improved version of the Trudinger-Moser inequality on the open unit disk $B\subset\R^2$, recently proved by G. Mancini and K. Sandeep. Unlike the original Trudinger-Moser inequality, this inequality is invariant with respect to Möbius automorphisms of the unit disk, and as such is a closer analogy of the critical nonlinearity $\int |u|^{2^*}$ in the higher dimension than the original Trudinger-Moser nonlinearity.

math.AP↗