We establish the area formula for change-of-variable mappings in the Sobolev space $W^{k,p}_{\text{loc}}$. Our approach relies on constructing Lipschitz approximations of Sobolev functions that agree with the original functions outside a set of Riesz capacity zero.
By developing an integral approach, we present a new method for the interior regularity of strictly convex solution of the Monge-Ampère equation $\det D^2 u = 1$.
An obsession with origins is a hallmark of pseudoarchaeology, while the celebration of arbitrary anniversaries is one of the more meaningless conceits of the heritage industry. In that spirit, I would like to wish a happy tenth anniversary to AP: Online Journal in Public Archaeology, and to extend my warmest congratulations to the editorial team.
Do we need a roadmap to the future? Or do we ‘wing it’, making it up as we go along? Big questions, but never more important than now, in this current time of uncertainty.Let’s start small, and refocus the question on our professional and scholarly area of interest and activities. While the future of the world of work certainly looks different – will robots do digging, recording and interpretation work in 2030?- I think that the key to prepare suitable strategies for going forward is to be clear about our purpose(s). For what, and for whom, are we and will we be doing research and knowledge sharing? With whom will we operate and work in our capacity as scholars, practitioners, teachers? Even asking why do archaeology may seem straightforward now, but it isn’t. At least, it shouldn’t be.
In this paper we prove the Liouville type theorem for stable at infinity solutions of the following equation $$Δ_{m}^{3}u =|u|^{θ-1}u\;\;\; \mbox{in}\,\, \mathbb{R}^N,$$ for $1<m-1<θ<θ_{s, m}:=\frac{N(m-1)+3m }{N-3m}.$ Here $θ_{s, m}$ is a the classic critical exponent for $m-$ bi-harmonic equation.
We obtain a new bound on the location of eigenvalues for a non-self-adjoint Schrödinger operator with complex-valued potentials by obtaining a weighted $L^2$ estimate for the resolvent of the Laplacian.
We obtain the global large solutions to the compressible Navier-Stokes equations in $\mathbb{R}^2$. The solution is large in the sense that there is no smallness assumption applied to one component of the initial incompressible velocity.
In this paper, we show the existence of a family of analytic stationary patch solutions of the SQG and gSQG equations. This answers an open problem in [F. de la Hoz, Z. Hassainia, T. Hmidi. Arch. Ration. Mech. Anal., 220(3):1209-1281, 2016].
These notes record and expand the lectures for the `Journées Équations aux Dérivées Partielles 2018' held by the author during the week of June 11-15, 2018. The aim is to give a overview of the classical theory for the obstacle problem, and then present some recent developments on the regularity of the free boundary.
In this note, we study the mapping properties of global pseudo-differential operators with symbols in Ruzhansky-Turunen classes on Besov spaces $B^{s}_{\infty,\infty}(G).$ The considered classes satisfy Fefferman type conditions of limited regularity.
We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-Δu=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.
Do you watch porn? Most people lie when answering this question.The pornographic industry would not be as big as it is if nobodyconsumed it. As with other cultural expressions, archaeology andthe past are also represented in porn movies, affecting the publicimage of our discipline as it does advertising, literature, or cinema.This paper explores and analyses the multiple references to thepast and our profession found within the context of pornographicmovies and other erotic products, highlighting the potential ofpornography as another tool for informal education.
In this paper, we study the singular set of 3-dimensional Navier-Stokes equations. Under the condition$\frac{1}{R^{\frac{3s}{q}+2-s}}\int^{R^{2}}_{0}(\int_{B_{R}}|u|^{q}dx)^{\frac{s}{q}}ds <C,$ for $(q,s)\in\{(2,5),(5,2)\},$ we use the backward uniqueness of parabolic equations to show that the Hausdorff dimension of the singular set is less than 1.
This paper is concerned with global existence of weak solutions for a weakly dissipative $μ$HS equation by using smooth approximate to initial data and Helly$^{,}$s theorem.
We consider an inverse problem arising in corrosion detection. We prove a stability result of logarithmic type for the determination of the corroded portion of the boundary and impedance by two measurements on the accessible portion of the boundary.
In this paper we study the asymptotic behavior of the solution of quasilinear parametric variational inequalities posed in a cylinder with a thin neck, and we obtain the limit problem.
New Hardy and Sobolev type inequalities involving $L^1$-norms of scalar and vector-valued functions in $\Bbb{R}^n$ are obtained. The work is related to some problems stated in the recent paper by Bourgain and Brezis