Waves of maximal height for a class of nonlocal equations with homogeneous symbols

We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order $-r$, where $r>1$. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol $m(k)=k^{-2}$. Thereby we recover its unique highest $2\pi$-periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest.Comment: 25 page

Existence and regularity theory in weighted Sobolev spaces and applications

In the thesis we discuss several questions related to the study of degenerate, possibly nonlinear PDEs of elliptic type. At first we discuss the equivalent conditions between the validity of weighted Poincar\'e inequalities, structure of the functionals on weighted Sobolev spaces, isoperimetric inequalities and the existence and uniqueness of solutions to the degenerate nonlinear elliptic PDEs with nonhomogeneous boundary condition, having the form:\begin{eqnarray}\label{eqn:abs}\left\{\begin{array}{lll}{\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) =x^*,\\~~~~~~~~~~~~u-w \in W^{1,p}_{\rho,0} (\Omega),\end{array}\right.\end{eqnarray}involving any given $x^*\in (W^{1,p}_{\rho,0} (\Omega))^*$ and $w\in W^{1,p}_{\rho} (\Omega)$, where $u\in W^{1,p}_{\rho} (\Omega)$ and $W^{1,p}_{\rho} (\Omega)$ denotes certain weighted Sobolev space, $W^{1,p}_{\rho,0} (\Omega)$ is the completion of $\mathcal{C}_{0}^{\infty}(\Omega)$. As a next step, we undertake a natural question how to interpret the nonhomogenous boundary conditions in weighted Sobolev spaces, when the natural analytical tools, like trace embedding theorems, are missing. Our further goal is to contribute to solvability and uniqueness for degenerate elliptic PDEs with nonhomogenous boundary condition being the extension of~\eqref{eqn:abs}. In addition to the monotonicity method used in the first step of our discussion for the problem~\eqref{eqn:abs}, we also exploit Lax-Miligram theorem to treat the linear problem like:\begin{equation*}\begin{cases}-{\rm div} (A(x)\nabla u(x)) + B(x)\cdot\nabla u(x) + C(x)u(x) = x^{*}\ \ \text{for a.e.}\ x\in \Omega, \\~~~~~~~~~~~~~~~~~~ u(x) = g(x) \ \ \text{for a.e. }\ x\in \partial\Omega ,\end{cases}\end{equation*}as well as Ekeland's Variational Principle and Boccardo-Murat techniques to consider problem like:\begin{align*} \begin{cases} - {\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) - \lambda\, b(x)| u|^{p-2} u = x^*,\\~~~~~~~~~~~~~~~~~~~u-z \in X , \end{cases}\end{align*}where $p>1,\ \lambda>0$, and the operator $\mathcal{L}_{\lambda} u:= - {\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) - \lambda\, b(x)| u|^{p-2} u$ is non-monotone.For the study of the nonhomogeneous BVPs, we apply recent results due to Ka\l{}amajska and myself, where we constructed trace extension operator from weighted Orlicz-Slobodetskii spaces defined on the boundary of the domain to weighted Orlicz-Sobolev spaces in the domain. Information on the spectrum of the corresponding differential operator is also derived. Moreover, some nonexistence and nonuniqueness results are also analyzed

Waves of maximal height for a class of nonlocal equations with homogeneous symbols

We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order -r, where r > 1. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol m(k) = k$^{-2}$. Thereby we recover its unique highest 2π-periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest

Formal methods for Multiscale models derivation

We are currently developing a software dedicated to multiscale and multiphysics model derivation oriented to arrays of micro and nanosystems. The software is based on the two-scale transform, together with formal specification and verification techniques in computer science. It helps to derive multiphysics models for complex geometries including thin or periodic structures or combinations of them. Final models are correct by construction since human errors are avoided and model derivation effort is dramatically reduced

Large solutions of degenerate and/or singular quasilinear elliptic equations in a ball

We consider local weak large solutions with its blow-up rate near the boundary to certain class of degenerate and/or singular quasilinear elliptic equation\\ ${\rm div}(d^{\alpha}(x,\partial{}B)\Phi_p(\nabla u)) = b(x)f(u)$ in a ball B, where $f$ is normalized regularly varying at infinity with index $\sigma+1>p-1,\ p>1$. In particular, how the asymptotic behavior of the solution changes over the varying index and degeneracy and/ or singularity present in the equation. We also include the second order blow-up rate for the corresponding semilinear problem

Nonradiality of second eigenfunctions of the fractional Laplacian in a ball

Using symmetrization techniques, we show that, for every N>=2, any second eigenfunction of the fractional Laplacian in the N-dimensional unit ball with homogeneous Dirichlet conditions is nonradial, and hence its nodal set is an equatorial section of the ball

Nonradiality of second eigenfunctions of the fractional Laplacian in a ball

Using symmetrization techniques, we show that, for every $N \geq 2$, any second eigenfunction of the fractional Laplacian in the $N$-dimensional unit ball with homogeneous Dirichlet conditions is nonradial, and hence its nodal set is an equatorial section of the ball.Comment: 13 pages, 2 figure

Rewriting Strategies for a Two-Scale Method: Application to Combined Thin and Periodic Structures

International audienceMultiphysics models of large arrays of micro- and nanosystems are too complex to be efficiently simulated by existing simulation software. Fortunately, asymptotic methods such as those based on two-scale convergence are applicable to homogenization of thin or periodic (i.e. array) structures. They generate simpler models tractable to simulation, but their application is long and requires a mathematical expertise. Our goal is to provide engineers with an implementation of this mathematical tool inside a modeling software.We follow therefore a multidisciplinary approach which combines a generalization and formalization effort of mathematical asymptotic methods, together with rewriting-based formal transformation techniques from computer science. This paper describes this approach, illustrates it with an example and presents the architecture of the software under design