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Vivek Tewary

Publications and source records attributed to Vivek Tewary.

At least 19 recordsLinked to original sources
arXiv2026

Equivalence of weak and viscosity solutions for nonlocal p-Laplace type equations on the Heisenberg group

We establish the equivalence between weak and viscosity solutions for a broad class of nonlocal $p$-Laplace type equations on the Heisenberg group whose kernels satisfy standard symmetry, ellipticity, and left-translation invariance assumptions. As a particular case, our results apply to the fractional Heisenberg $p$-Laplacian. The proof combines intrinsic approximation by modified infimal convolutions, Heisenberg mollification, and a nonlocal integration-by-parts argument adapted to the sub-Riemannian geometry. The main analytical difficulty stems from the noncommutative structure of the Heisenberg group, which prevents a direct extension of the Euclidean theory and requires new localization and approximation techniques.

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arXiv2025

Nonlocal Quasilinear Parabolic Equations in Heisenberg Group: Local Boundedness with an Optimal Tail

We prove local boundedness for a quasilinear parabolic equation on the Heisenberg group \[ \partial_t u(\xi,t) + \text{p.v.}\int_{\mathbb{H}^N} \frac{|u(\xi,t)-u(\eta,t)|^{p-2}(u(\xi,t)-u(\eta,t))}{|\eta^{-1}\circ \xi|^{Q+sp}} \,d\eta = 0, \] with optimal regularity assumption on the tail term. We also prove interpolation inequalities and an extension theorem for fractional Sobolev spaces on the Heisenberg group.

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arXiv2024

A variational approach to nonlocal image restoration flows

We prove existence, uniqueness and initial time regularity for variational solutions to nonlocal total variation flows associated with image denoising and deblurring. In particular, we prove existence of parabolic minimisers $u$, that is, $$\int_0^T\int_\Omega u\partial_t\phi\,dx + \textbf{F}(u(t))\,dt\leq \int_0^T \textbf{F}(u+\phi)(t)\,dt,$$ for $\phi\in C^\infty_c(\Omega\times (0,T))$. The prototypical functional $\textbf{F}(u)$ is $\textbf{F}(u)=\textbf{TV}^{\alpha}_{\cdot}(u)+\frac{\kappa}{\zeta}\int_\Omega|u(x)-u_0(x)|^\zeta\,dx$ for $\zeta\geq 1$. Here $\textbf{TV}^{\alpha}_{\cdot}$ is a fractional total variation of either the Riesz or the Gagliardo type and the second term is a regression term. These models are based on different definitions of fractional $\textbf{BV}$ spaces that have been proposed in the literature. The notion of solution is completely variational and based on the weighted dissipation method. We demonstrate existence without smoothness assumptions on the domain and exhibit uniqueness without using strict convexity. We can also deal with fairly general fidelity or regression terms in the model. Furthermore, the method also provides a novel route to constructing solutions of the parabolic fractional $1$-Laplace equation.

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arXiv2023

Gradient regularity for mixed local-nonlocal quasilinear parabolic equations

In this paper, we prove local H\"older continuity for the spatial gradient of weak solutions to $$u_t - \text{div} (|\nabla u|^{p-2}\nabla u) + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+ps}} \ dy = 0.$$ It is easy to see that parabolic quasilinear equations are not scaling invariant and this led to the development of the method of intrinsic scaling by E.DiBenedetto, E.DiBenedetto-Y.Z.Chen, J.Kinnunen-J.Lewis and A.Friedman-E.DiBenedetto. In a very recent paper, C.de Filippis-G.Mingione proved gradient H\"older continuity for mixed local-nonlocal quasilinear elliptic equations and in this paper, we extend this result to the parabolic case. Since we only expect regularity for $\nabla_x u$ in the parabolic setting, it is not clear how to extend the elliptic proof to the parabolic case. In order to overcome this difficulty, we instead follow the ideas developed by T.Kuusi-G.Mingione combined with the novel tail estimates of C.deFilippis-G.Mingione. An advantage of our approach is that we can obtain both $C^{1,\alpha}_x$ regularity as well as $C^{0,1} _x$ potential estimates in one go. Moreover, we do not need to make use of any form of Caccioppoli inequality and instead, the regularity is obtained only through a suitable difference estimate.

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arXiv2022

Hölder regularity for fractional $p$-Laplace equations

We give an alternative proof for Hölder regularity for weak solutions of nonlocal elliptic quasilinear equations modelled on the fractional p-Laplacian where we replace the discrete De Giorgi iteration on a sequence of concentric balls by a continuous iteration. This work can be viewed as the nonlocal counterpart to the ideas developed by Tiziano Granucci.

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arXiv2022

$C^{1,α}$ regularity for quasilinear parabolic equations with nonstandard growth

In this paper, we obtain $C^{1,α}$ estimates for weak solutions of certain quasilinear parabolic equations satisfying nonstandard growth conditions, the prototype examples being $$u_t - \text{div} (|\nabla u|^{p-2} \nabla u + a(t)|\nabla u|^{q-2} \nabla u) = 0,$$ $$u_t - \text{div} (|\nabla u|^{p(t)-2} \nabla u) = 0.$$ under the assumption that the solutions a priori have bounded gradient. We build on the recently developed scaling and covering argument which allows us to consider the singular and degenerate cases in a uniform manner and with minimal regularity requirements on the phase switching factor $a(t)$ and the variable exponent $p(t)$. Moreover, we are able to take any $p \leq q < \infty$ to obtain the desired regularity.

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arXiv2022

Pointwise and Weighted Hessian Estimates for Kolmogorov-Fokker-Planck type operators

In this article, we obtain hessian estimates for Kolmogorov-Fokker-Planck operators in non-divergence form in several Banach function spaces. Our approach relies on a representation formula and newly developed sparse domination techniques in Harmonic Analysis. Our result when restricted to weighted Lebesgue spaces yields sharp quantitative hessian estimates for the Kolmogorov-Fokker-Planck operators.

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arXiv2022

Borderline Lipschitz regularity for bounded minimizers of functionals with (p,q)-growth

We prove local Lipschitz regularity for bounded minimizers of functionals with nonstandard $p,q$-growth with the source term in the Lorentz space $L(N,1)$ under the restriction $q<p+1+p\,\min\left\{\frac 1N,\frac{2(p-1)}{Np-2p+2}\right\}$. This extends the recent work by Beck-Mingione to bounded minimizers under weaker hypothesis and is sharp for some special ranges of $p$, $q$ and $N$.

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arXiv2022

Local boundedness of variational solutions to nonlocal double phase parabolic equations

We prove local boundedness of variational solutions to the double phase equation \begin{align*} \partial_t u +& P.V.\int_{\mathbb{R}^N}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\ &+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qs'}} \,dy = 0, \end{align*} under the restrictions $s,s'\in (0,1),\, 1 < p \leq q \leq p\,\frac{2s+N}{N}$ and the non-negative function $(x,y)\mapsto a(x,y)$ is assumed to be measurable and bounded.

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arXiv2022

Existence of variational solutions to nonlocal evolution equations via convex minimization

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u &+ \text{P.V.}\int_{\mathbb{R}^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\&+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qr}} \,dy = 0. \end{align*} The approach of minimization of parameter-dependent convex functionals over space-time trajectories requires only appropriate convexity and coercivity assumptions on the nonlocal operator. As the parameter tends to zero, we recover variational solutions. Under further growth conditions, these variational solutions are global weak solutions. Further, this provides a direct minimization approach to approximation of nonlocal evolution equations.

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arXiv2022

Existence of variational solutions to doubly nonlinear nonlocal evolution equations via minimizing movements

We prove existence of variational solutions for a class of doubly nonlinear nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u^m &+ \text{P.V.}\int_{\mathbb{R}^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\&+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qr}} \,dy = 0,\,m>0,\,p>1,\,s,r\in (0,1). \end{align*} We make use of the approach of minimizing movements pioneered by DeGiorgi and Ambrosio and refined by Bögelein, Duzaar, Marcellini, and co-authors to study nonlinear parabolic equations with non-standard growth.

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arXiv2021

On Lipschitz regularity for bounded minimizers of functionals with (p,q) growth

We obtain Lipschitz estimates for bounded minimizers of functionals with nonstandard $(p,q)$-growth satisfying the dimension-independent restriction $q \frac{p(2+p)}{2} + 1$. The standard Lipschitz regularity takes the form $W^{1,\infty}_{\text{loc}} - W^{1,p}_{\text{loc}}$, whereas we obtain $W^{1,\infty}_{\text{loc}} - L^{\infty}_{\text{loc}}$ regularity estimate and then make use of existing sharp $L^{\infty}_{\text{loc}}$ bounds to obtain the required conclusion.

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arXiv2020

Combined Effects of Homogenization and Singular Perturbations : A Bloch Wave Approach

We study Bloch wave homogenization of periodically heterogeneous media with fourth order singular perturbations. We recover different homogenization regimes depending on the relative strength of the singular perturbation and length scale of the periodic heterogeneity. The homogenized tensor is obtained in terms of the first Bloch eigenvalue. The higher Bloch modes do not contribute to the homogenization limit.

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arXiv2020

Bloch wave approach to almost periodic homogenization and approximations of effective coefficients

Bloch wave homogenization is a spectral method for obtaining effective coefficients for periodically heterogeneous media. This method hinges on the direct integral decomposition of periodic operators, which is not available in a suitable form for almost periodic operators. In particular, the notion of Bloch eigenvalues and eigenvectors does not exist for almost periodic operators. However, we are able to recover the homogenization result in this case, by employing a sequence of periodic approximations to almost periodic operators. We also establish a rate of convergence for approximations of homogenized tensors for a class of almost periodic media. The results are supported by a numerical study.

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arXiv2019

Bloch Wave Homogenization of Quasiperiodic Media

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a "cut and project" procedure. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift an almost periodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularized version of the degenerate equation. Homogenized coefficients for quasiperiodic media are determined in terms of the first Bloch eigenvalue of the regularized lifted equation. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenization limit for an equation with highly oscillating quasiperiodic coefficients.

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arXiv2019

Generic Simplicity of Spectral Edges and Applications to Homogenization

We consider the spectrum of a second-order elliptic operator in divergence form with periodic coefficients, which is known to be completely described by Bloch eigenvalues. We show that under small perturbations of the coefficients, a multiple Bloch eigenvalue can be made simple. The Bloch wave method of homogenization relies on the regularity of spectral edge. The spectral tools that we develop, allow us to obtain simplicity of an internal spectral edge through perturbation of the coefficients. As a consequence, we are able to establish Bloch wave homogenization at an internal edge in the presence of multiplicity by employing the perturbed Bloch eigenvalues. We show that all the crossing Bloch modes contribute to the homogenization at the internal edge and that higher and lower modes do not contribute to the homogenization process.

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