Mathematics - Analysis of PDEs Publications (50)


Mathematics - Analysis of PDEs Publications

In this paper, we will study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $(u,v)$ from a smooth bounded open domain $\Omega\subset\R^m$ to a Lorentzian manifold with Dirichlet boundary condition, we prove that it is smooth outside a closed set whose $(m-2)$-dimension Hausdorff measure is zero. Moreover, if the target manifold $N$ does not admit any harmonic sphere $S^l$, $l=2,. Read More

We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schr\"odinger operator $ L_{\gamma,\alpha}: = ({-}{ \Delta})^{\frac{\alpha}{2}}- \frac{\gamma}{|x|^{\alpha}}$ on domains of $\mathbb{R}^n$ containing the singularity $0$, where $0<\alpha<2$ and $ 0 \le \gamma < \gamma_H(\alpha)$, the latter being the best constant in the fractional Hardy inequality on $\mathbb{R}^n$. We tackle the existence of least-energy solutions for the borderline boundary value problem $(L_{\gamma,\alpha}-\lambda I)u= {\frac{u^{2^\star_\alpha(s)-1}}{|x|^s}}$ on $\Omega$, where $0\leq s <\alpha Read More

We develop an asymptotical control theory for one of the simplest distributed oscillating system --- the closed string under a bounded load applied to a single distinguished point. We find exact classes of the string states that allows complete damping, and asymptotically exact value of the required time. By using approximate reachable sets instead of exact ones we design a dry-friction like feedback control, which turns to be asymptotically optimal. Read More

We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation, the Benjamin-Ono equation), the nonlinear Schr\"odinger equation and the BBM equation. Read More

We study invasion fronts and spreading speeds in two component reaction-diffusion systems. Using Lin's method, we construct traveling front solutions and show the existence of a bifurcation to locked fronts where both components invade at the same speed. Expansions of the wave speed as a function of the diffusion constant of one species are obtained. Read More

Traveling periodic waves of the modified Korteweg-de Vries (mKdV) equation are considered in the focusing case. By using one-fold and two-fold Darboux transformations, we construct explicitly the rogue periodic waves of the mKdV equation expressed by the Jacobian elliptic functions dn and cn respectively. The rogue dn-periodic wave describes propagation of an algebraically decaying soliton over the dn-periodic wave, the latter wave is modulationally stable with respect to long-wave perturbations. Read More

We consider the semilinear heat equation \begin{eqnarray*} \partial_t u = \Delta u + |u|^{p-1} u \ln ^{\alpha}( u^2 +2), \end{eqnarray*} in the whole space $\mathbb{R}^n$, where $p > 1$ and $ \alpha \in \mathbb{R}$. Unlike the standard case $\alpha = 0$, this equation is not scaling invariant. We construct for this equation a solution which blows up in finite time $T$ only at one blowup point $a$, according to the following asymptotic dynamics: \begin{eqnarray*} u(x,t) \sim \psi(t) \left(1 + \frac{(p-1)|x-a|^2}{4p(T -t)|\ln(T -t)|} \right)^{-\frac{1}{p-1}} \text{ as } t \to T, \end{eqnarray*} where $\psi(t)$ is the unique positive solution of the ODE \begin{eqnarray*} \psi' = \psi^p \ln^{\alpha}(\psi^2 +2), \quad \lim_{t\to T}\psi(t) = + \infty. Read More

We investigate the initial-boundary value problem for the general three-component nonlinear Schrodinger (gtc-NLS) equation with a 4x4 Lax pair on a finite interval by extending the Fokas unified approach. The solutions of the gtc-NLS equation can be expressed in terms of the solutions of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Moreover, the relevant jump matrices of the RH problem can be explicitly found via the three spectral functions arising from the initial data, the Dirichlet-Neumann boundary data. Read More

The Boutet de Monvel calculus of pseudo-differential boundary operators is generalised to the full scales of Besov and Triebel--Lizorkin spaces (though with finite integral exponents for the latter). The continuity and Fredholm properties proved here extend those previously obtained by Franke and Grubb, and the results on range complements of surjectively elliptic Green operators improve the earlier known, even for the classical spaces with $1Read More

We investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii (GP) equations with a 4x4 Lax pair on the half-line. The solution of this system can be obtained in terms of the solution of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be explicitly found using the two spectral functions s(k) and S(k), which can be defined by the initial data, the Dirichlet-Neumann boundary data at x=0. Read More

The dynamics along the particle trajectories for the 3D axisymmetric Euler equations are considered. It is shown that if the inflow is rapidly increasing (pushy) in time, the corresponding laminar profile of the incompressible Euler flow is not (in some sense) stable provided that the swirling component is not zero. It is also shown that if the vorticity on the axis is not zero (with some extra assumptions), then there is no steady flow. Read More

We prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the initial data has slope strictly less than 1. The curvature of these solutions solutions decays to 0 as $t$ goes to infinity, and they are unique when the initial data is $C^{1,\epsilon}$. We do this by constructing a modulus of continuity generated by the equation, just as Kiselev, Nazarov, and Volberg did in their proof of the global well-posedness for the quasi-geostraphic equation. Read More

We consider the problem of a one dimensional elastic filament immersed in a two dimensional steady Stokes fluid. Immersed boundary problems in which a thin elastic structure interacts with a surrounding fluid are prevalent in science and engineering, a class of problems for which Peskin has made pioneering contributions. Using boundary integrals, we first reduce the fluid equations to an evolution equation solely for the immersed filament configuration. Read More

About 15 years ago, Bismut gave a natural construction of a Hodge theory for a hypoelliptic Laplacian acting on the total space of the cotangent bundle of a Riemannian manifold. This operator interpolates between the classical elliptic Laplacian on the base and the generator of the geodesic flow. We will describe recent developments of the theory of hypoelliptic Laplacians, in particular the explicit formula obtained by Bismut for orbital integrals and the recent solution by Shen of Fried's conjecture (dating back to 1986) for locally symmetric spaces. Read More

This article extends the author's past work [Inv. Probl. Imaging, 10:2 (2016), 433--459] to attenuated X-ray transforms, where the attenuation is complex-valued and only depends on position. Read More

We consider a haptotaxis cancer invasion model that includes two families of cancer cells. Both families, migrate on the extracellular matrix and proliferate. Moreover the model describes an epithelial-to-mesenchymal-like transition between the two families, as well as a degradation and a self-reconstruction process of the extracellular matrix. Read More

In this paper we study a semilinear problem for the fractional laplacian that are the counterpart of the Neumann problems in the classical setting. We show uniqueness of minimal energy solutions for small domains. Read More

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Read More

This paper investigates the asymptotic behaviour of solutions of periodic evolution equations. Starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class of dissipative systems arising naturally in applications. For this class of systems we analyse in detail the spectral properties of the associated monodromy operator, showing in particular that it is a so-called Ritt operator under a natural 'resonance' condition. Read More

We show that a sufficient condition for the weak limit of a sequence of $W^1_q$-homeomorphisms with finite distortion to be almost everywhere injective for $q \geq n-1$, can be stated by means of composition operators. Applying this result, we study nonlinear elasticity problems with respect to these new classes of mappings. Furthermore, we impose loose growth conditions on the stored-energy function for the class of $W^1_n$-homeomorphisms with finite distortion and integrable inner as well as outer distortion coefficients. Read More

In this paper, we investigate an optimal design problem motivated by some issues arising in population dynamics. In a nutshell, we aim at determining the optimal shape of a region occupied by resources for maximizing the survival ability of a species in a given box and we consider the general case of Robin boundary conditions on its boundary. Mathematically, this issue can be modeled with the help of an extremal indefinite weight linear eigenvalue problem. Read More

We study a nonlinear boundary value problem driven by the $p$-Laplacian plus an indefinite potential with Robin boundary condition. The reaction term is a Carath\'eodory function which is asymptotically resonant at $\pm\infty$ with respect to a nonprincipal Ljusternik-Schnirelmann eigenvalue. Using variational methods, together with Morse theory and truncation-perturbation techniques, we show that the problem has at least three nontrivial smooth solutions, two of which have a fixed sign. Read More

Let $\L$ be a Schr\"odinger operator of the form $\L=-\Delta+V$ acting on $L^2(\mathbb R^n)$, $n\geq3$, where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ Let ${\rm BMO}_{{\mathcal{L}}}(\RR)$ denote the BMO space associated to the Schr\"odinger operator $\L$ on $\RR$. In this article we show that for every $f\in {\rm BMO}_{\mathcal{L}}(\RR)$ with compact support, then there exist $g\in L^{\infty}(\RR)$ and a finite Carleson measure $\mu$ such that $$ f(x)=g(x) + S_{\mu, {\mathcal P}}(x) $$ with $\|g\|_{\infty} +\||\mu\||_{c}\leq C \|f\|_{{\rm BMO}_{\mathcal{L}}(\RR)},$ where $$ S_{\mu, {\mathcal P}}=\int_{{\mathbb R}^{n+1}_+} {\mathcal P}_t(x,y) d\mu(y, t), $$ and ${\mathcal P}_t(x,y)$ is the kernel of the Poisson semigroup $\{e^{-t\sqrt{\L}}\}_{t> 0} $ on $L^2(\mathbb R^n)$. Read More

We obtain necessary conditions and sufficient conditions for the solvability of the heat equation in a half-space of ${\bf R}^N$ with a nonlinear boundary condition. Furthermore, we study the relationship between the life span of the solution and the behavior of the initial function. Read More

This paper is concerned with the theoretical study of plasmonic resonances for linear elasticity governed by the Lam\'e system in $\mathbb{R}^3$, and their application for cloaking due to anomalous localized resonances. We derive a very general and novel class of elastic structures that can induce plasmonic resonances. It is shown that if either one of the two convexity conditions on the Lam\'e parameters is broken, then we can construct certain plasmon structures that induce resonances. Read More

We derive a continuum model for incompatible elasticity as a variational limit of a family of discrete nearest-neighbor elastic models. The discrete models are based on discretizations of a smooth Riemannian manifold $(M,\mathfrak{g})$, endowed with a flat, symmetric connection $\nabla$. The metric $\mathfrak{g}$ determines local equilibrium distances between neighboring points; the connection $\nabla$ induces a lattice structure shared by all the discrete models. Read More

The study of Einstein constraint equations in general relativity naturally leads to considering Riemannian manifolds equipped with nonsmooth metrics. There are several important differential operators on Riemannian manifolds whose definitions depend on the metric: gradient, divergence, Laplacian, covariant derivative, conformal Killing operator, and vector Laplacian, among others. In this article, we study the approximation of such operators, defined using a rough metric, by the corresponding operators defined using a smooth metric. Read More

In the Fermi Lectures on the obstacle problem in 1998, Caffarelli gave a proof of the mean value theorem which extends to general divergence form uniformly elliptic operators. In the general setting, the result shows that for any such operator $L$ and at any point $x_0$ in the domain, there exists a nested family of sets $\{ D_r(x_0) \}$ where the average over any of those sets is related to the value of the function at $x_0.$ Although it is known that the $\{ D_r(x_0) \}$ are nested and are comparable to balls in the sense that there exists $c, C$ depending only on $L$ such that $B_{cr}(x_0) \subset D_r(x_0) \subset B_{Cr}(x_0)$ for all $r > 0$ and $x_0$ in the domain, otherwise their geometric and topological properties are largely unknown. Read More

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension $8$, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin's theorem is optimal. Read More

We prove a compactness principle for the anisotropic formulation of the Plateau problem in any codimension, in the same spirit of the previous works of the authors \cite{DelGhiMag,DePDeRGhi,DeLDeRGhi16}. In particular, we perform a new strategy for the proof of the rectifiability of the minimal set, based on the new anisotropic counterpart of the Allard rectifiability theorem proved by the authors in \cite{DePDeRGhi2}. As a consequence we provide a new proof of Reifenberg existence theorem. Read More

The combined work of Guaraco, Hutchinson, Tonegawa and Wickramasekera has recently produced a new proof of the classical theorem that any closed Riemannian manifold of dimension $n + 1 \geq 3$ contains a minimal hypersurface with a singular set of Hausdorff dimension at most $n-7$. This proof avoids the Almgren--Pitts geometric min-max procedure for the area functional that was instrumental in the original proof, and is instead based on a considerably simpler PDE min-max construction of critical points of the Allen--Cahn functional. Here we prove a spectral lower bound for the hypersurfaces arising from this construction. Read More

In this paper, we study the asymptotic behavior of solutions to the wave equation with damping depending on the space variable and growing at the spatial infinity. We prove that the solution is approximated by that of the corresponding heat equation as time tends to infinity. The proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation. Read More

Bounded weak solutions of Burgers' equation $\partial_tu+\partial_x(u^2/2)=0$ that are not entropy solutions need in general not be $BV$. Nevertheless it is known that solutions with finite entropy productions have a $BV$-like structure: a rectifiable jump set of dimension one can be identified, outside which $u$ has vanishing mean oscillation at all points. But it is not known whether all points outside this jump set are Lebesgue points, as they would be for $BV$ solutions. Read More

This paper proposes the use of the Spectral method to simulate diffusive moisture transfer through porous materials, which can be strongly nonlinear and can significantly affect sensible and latent heat transfer. An alternative way for computing solutions by considering a separated representation is presented, which can be applied to both linear and nonlinear diffusive problems, considering highly moisture-dependent properties. The Spectral method is compared with the classical implicit Euler and Crank-Nicolson schemes. Read More

This note revolves on the free Dirac operator in $\mathbb{R}^3$ and its $\delta$-shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by finding sharp constants and minimizers of some precise inequalities related to an uncertainty principle. On the other hand, we prove that the domains given by Dittrich, Exner and \v{S}eba [Dirac operators with a spherically symmetric $\delta$-shell interaction, J. Read More

We prove, under some assumptions, the existence of correctors for the stochastic homoge-nization of of " viscous " possibly degenerate Hamilton-Jacobi equations in stationary ergodic media. The general claim is that, assuming knowledge of homogenization in probability, correctors exist for all extreme points of the convex hull of the sublevel sets of the effective Hamiltonian. Even when homogenization is not a priori known, the arguments imply existence of correctors and, hence, homogenization in some new settings. Read More

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with Dirichlet boundary condition on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. The key tool consists in combining classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic equation. Read More

We prove the $W^{2s,p}_{\textrm{loc}}$ local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of $\mathbb{R}^N$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. Read More

We develop some of the basic theory for the obstacle problem on Riemannian Manifolds, and we use it to establish a mean value theorem. Our mean value theorem works for a very wide class of Riemannian manifolds and has no weights at all within the integral. Read More

In this paper, we will discuss the use of a Sampling Method to reconstruct impenetrable inclusions from Electrostatic Cauchy data. We consider the case of a perfectly conducting and impedance inclusion. In either case, we show that the Dirichlet to Neumann mapping can be used to reconstruct impenetrable sub-regions via a sampling method. Read More

In this paper, we prove the existence of global weak solutions to the compressible Navier-Stokes equations when the pressure law is in two variables.The method is based on the Lions argument and the Feireisl-Novotny-Petzeltova method. The main contribution of this paper is to develop a new argument for handling a nonlinear pressure law $P(\rho,n)=\rho^{\gamma}+n^{\alpha}$ where $\rho,\,n$ satisfy the mass equations. Read More

We prove Gaussian upper and lower bounds for the fundamental solutions of a class of degenerate parabolic equations satisfying a weak Hormander condition. The bound is independent of the smoothness of the coefficients and generalizes classical results for uniformly parabolic equations Read More

This paper is concerned with a viscoelastic equation of Kirchhoff type with acoustic boundary conditions in a bounded domain of $\mathbb{R}^{n}.$ We show that, under suitable conditions on the initial data, the solution exists globally in time. Then, we prove the general energy decay of global solutions by applying a lemma of P. Read More

In this paper we express tau functions for the Korteweg de Vries (KdV) equation, as Laplace transforms of iterated Skorohod integrals. Our main tool is the notion of Fredholm determinant of an integral operator. Our result extends the paper of Ikeda and Taniguchi who obtained a stochastic representation of tau functions for the $N$-soliton solutions of KdV as the Laplace transform of a quadratic functional of $N$ independent Ornstein-Uhlenbeck processes. Read More

We show that the maximal Cheeger set of a Jordan domain $\Omega$ without necks is the union of all balls of radius $r = h(\Omega)^{-1}$ contained in $\Omega$. Here, $h(\Omega)$ denotes the Cheeger constant of $\Omega$, that is, the infimum of the ratio of perimeter over area among subsets of $\Omega$, and a Cheeger set is a set attaining the infimum. The radius $r$ is shown to be the unique number such that the area of the inner parallel set $\Omega^r$ is equal to $\pi r^2$. Read More

Let $L_g$ be the subcritical GJMS operator on an even-dimensional compact manifold $(X, g)$ and consider the zeta-regularized trace $\mathrm{Tr}_\zeta(L_g^{-1})$ of its inverse. We show that if $\ker L_g = 0$, then the supremum of this quantity, taken over all metrics $g$ of fixed volume in the conformal class, is always greater than or equal to the corresponding quantity on the standard sphere. Moreover, we show that in the case that it is strictly larger, the supremum is attained by a metric of constant mass. Read More

We consider forward-forward Mean Field Game (MFG) models that arise in numerical approximations of stationary MFGs. First, we establish a link between these models and a class of hyperbolic conservation laws as well as certain nonlinear wave equations. Second, we investigate existence and long-time behavior of solutions for such models. Read More

We prove a priori estimates for real-valued periodic solutions to the modified Benjamin-Ono equation for initial data in $H^s$ where s>1/4. Our approach relies on localizing Fourier restriction spaces in time, after which one recovers the dispersive properties from Euclidean space. Read More

Three problems about recovery of a high-frequency free term in the one-dimension wave equation with homogeneous initial-boundary conditions by some information about partial asymptotics of its solution have been solved. It is shoun, that the free term can be completely recovered from a specific data about incomplete (three-terms) asymptotics of the solution. Before formulation of the each problem about recovery of free term, construction and justification of the asymptotics of the solution of original initial-boundary problem are given. Read More

Semi-linear elliptic boundary problems with non-linearities of product type are considered, in particular the stationary Navier--Stokes equations. Regularity and existence results are dealt with in the Besov and Triebel--Lizorkin spaces, and it is explained how difficulties occurring for boundary conditions of a high class may be handled. Read More