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Ill posed linear and nonlinear initial value problems may be stabilized, that it converted to to well posed initial value problems, by the addition of purely nonscalar linear dispersive terms. This is a stability analog of the Turing instability. |
Let $A = - \sum \partial_k \, c_{kl} \, \partial_l$ be a degenerate sectorial differential operator with complex bounded mesaurable coefficients. Let $\Omega \subset \mathds{R}^d$ be open and suppose that $A$ is strongly elliptic on $\Omega$. |
In this paper, we study the steady-states of a large class of stationary radiative transfer equations in a $C^1$ convex bounded domain. Namely, we consider the case in which both absorption-emission and scattering coefficients depend on the local temperature $T$ and the radiation frequency $\nu. |
It is known that the famous theoretical formula by Koiter for the critical buckling load of circular cylindrical shells under axial compression does not coincide with the experimental data. Namely, while Koiter's formula predicts linear dependence of the buckling load $\lambda(h)$ of the shell thickness $h$ ($h>... |
We prove $L^p$-mass concentration properties of Laplace eigenfunctions away from their nodal sets, extending a recent result in \cite{GM3} to all dimensions, and giving a slight refinement of a result in \cite{JN}. As a consequence, we are able to derive a sharp Wasserstein uncertainty principle that holds uniformly i... |
As is well-known, numerical experiments show that Napoleon's Theorem for planar triangles does not extend to a similar statement for triangles on the unit sphere $S^2$. Spherical triangles for which an extension of Napoleon's Theorem holds are called ``Napoleonic'', and until now the only known example... |
The Cauchy problem for the Klein-Gordon equation under the quartic potential is considered in the de Sitter spacetime. The existence of the global solution is shown based on the mechanism of the spontaneous symmetry breaking for the small positive Hubble constant. |
We discuss the inverse problem of determining the possible presence of an (n-1)-dimensional crack \Sigma in an n-dimensional body \Omega with n > 2 when the so-called Dirichlet-to-Neumann map is given on the boundary of \Omega. In combination with quantitative unique continuation techniques, an optimal single-logar... |
In this paper, we study the existence of normalized solutions for the nonautonomous Schrödinger-Poisson equations \begin{equation}\nonumber -\Delta u+\lambda u +\left(\vert x \vert ^{-1} * \vert u \vert ^{2} \right) u=A(x)|u|^{p-2}u,\quad \text{in}~\R^3, \end{equation} where $\lambda\in\R$, $A \in L^\infty(\R^3)$ satis... |
The study of spherically symmetric motion is important for the theory of explosion waves. In this paper, we construct rigorously self-similar solutions to the Riemann problem of the spherically symmetric Euler equations for general equations of state. |
We prove sharp Harnack inequalities for a family of Kolmogorov-Fokker-Planck type hypoelliptic diffusions. |
Large time dynamics of reaction-diffusion systems modeling some irreversible reaction networks are investigated. Depending on initial masses, these networks possibly possess boundary equilibria, where some of the chemical concentrations are completely used up. |
The motion of a rigid body immersed in an incompressible perfect fluid which occupies a three- dimensional bounded domain have been recently studied under its PDE formulation. In particular classical solutions have been shown to exist locally in time. |
We introduce a new method for analyzing nonlinear wave-Klein-Gordon systems and establishing global-in-time existence results for the Cauchy problem when the initial data need not have compact support. This method, which we call the Euclidian-Hyperboidal Foliation Method (EHFM), relies on the construction of a spaceti... |
We consider odd symmetric (1+1)-scalar field models with one internal mode. Under natural and robust assumptions, including the Fermi golden rule, we prove the asymptotic stability of the kink by odd perturbations in the energy space. |
This work is devoted to the analysis of the Yamabe problem on Spin manifolds and some applications to CMC immersions. Despite the efforts of many authors, very little is known on the existence of Yamabe metrics on general Spin manifolds. |
In this article we study the notion of capacity of a vertex for infinite graphs over non-Archimedean fields. In contrast to graphs over the real field monotone limits do not need to exist. |
We prove that the Harnack inequality fails for nonlocal kinetic equations. Such equations arise as linearized models for the Boltzmann equation without cutoff and are of hypoelliptic type. |
In this paper we consider the minimization of the functional \[ J[u]:=\int_\Omega |\Delta u|^2+\chi_{\{u>0\}} \] in the admissible class of functions \[ \mathcal A:= \left\{u\in W^{2, 2}(\Omega) {\mbox{ s.t. }} u-u_0\in W^{1,2}_0(\Omega) \right\}. \] Here, $\Omega$ is a smooth and bounded domain and $u_0\in W^{2,2}... |
We determine the large-time behavior of unbounded solutions for the so-called viscous Hamilton Jacobi equation, $u_t - \Delta u + |Du|^m = f(x)$, in the quadratic and subquadratic cases (i.e., for $1<m\leq 2$), with a particular focus on allowing arbitrary growth at infinity for $f$ and the prescribed initial data. ... |
We consider second-order elliptic equations in a half space with leading coefficients measurable in a tangential direction. We prove the $W^2_p$-estimate and solvability for the Dirichlet problem when $p\in (1,2]$, and for the Neumann problem when $p\in [2,\infty)$. |
These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. We emphasize the connexion between optimal constants and spec... |
We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid's viscosity in the mathematical model, we suggest to directly use experimental data. |
For Laplace-Beltrami operators associated to metrics which are long range perturbations of the flat one, we prove estimates for powers of the resolvent as the spectral parameter goes to zero. We also discuss applications to the local energy decay for the Schroedinger, Wave and Klein Gordon equations. |
We prove Holder regularity for solutions of non divergence integro-differential equations with non necessarily even kernels. The even/odd decomposition of the kernel can be understood as a sum of a diffusion and a drift term. |
Let $\Omega\subset\mathbb R^N$ be a bounded open set with Lipschitz continuous boundary $\Gamma$. Let $\gamma>0$, $\delta\ge 0$ be real numbers and $\beta$ a nonnegative measurable function in $L^\infty(\Gamma)$. |
We prove the existence of infinitely many classical periodic solutions for a class of degenerate semilinear wave equations: \[ u_{tt}-u_{xx}+|u|^{s-1}u=f(x,t), \] for all $s>1$. In particular we prove the existence of infinitely many classical solutions for the case $s=3$ posed by Brézis in \cite{BrezisBAMS}. |
A new numerical method to solve an inverse source problem for the Helmholtz equation in inhomogenous media is proposed. This method reduces the original inverse problem to a boundary value problem for a coupled system of elliptic PDEs, in which the unknown source function is not involved. |
We uncover a seemingly previously unnoticed algebraic structure of a large class of reaction-diffusion equations and use it, in particular, to study the long time behavior of the solutions and their convergence to traveling waves in the pulled and pushed regimes, as well as at the pushmi-pullyu boundary. One such new ... |
We prove Li-Yau and Harnack inequalities for systems of linear reaction-diffusion equations. By introducing an additional discrete spatial variable, the system is rewritten as a scalar diffusion equation with an operator sum. |
We consider a parabolic equation in a bounded domain $\OOO$ over a time interval $(0,T)$ with the homogeneous Neumann boundary condition. We arbitrarily choose a subboundary $\Gamma \subset \ppp\OOO$. |
We present two generalizations of the classical Stampacchia Lemma which contain a non-decreasing non-negative function $g$, and give applications. As a first application, we deal with variational integrals of the form $$ {\cal J} (u;\Omega) = \int_{\Omega}\ f(x,Du{(x)})dx. |
In this paper, we establish the existence and uniqueness theorem for entire solutions of Hessian equations with prescribed asymptotic behavior at infinity. This extends the previous results on Monge-Ampère equations. |
This paper proposes an abstract theory concerned with dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized se... |
We investigate the singular limit, as $\ep \to 0$, of the Fisher equation $\partial_t u=\ep \Delta u + \ep ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as $\Vert x \Vert \to \infty$. |
Reaction-diffusion equations are widely used to describe a variety of phenomena such as pattern formation and front propagation in biological, chemical and physical systems. In the one-dimensional model with a balanced bistable reaction function, it is well-known that there is persistence of metastable patterns for an... |
In this paper, we study the generation of eigenvalues of a stable monotonic shear flow under perturbations in $C^s$ with $s<2$. More precisely, we study the Rayleigh operator $\mathcal{L}_{U_{m,\gamma}}= U_{m,\gamma}\partial_x-U''_{m,\gamma}\partial_x\Delta^{-1}$ associated with perturbed shear flow $(U_{m,... |
We study degenerate Sobolev spaces where the degeneracy is controlled by a matrix $A_p$ weight. This class of weights was introduced by Nazarov, Treil and Volberg, and degenerate Sobolev spaces with matrix weights have been considered by several authors for their applications to PDEs. |
Inspired by Frehse's [1] 1973 work, we show that his elliptic system $\Delta u = F(u, \nabla u)$ in the plane has bounded weak solutions $u$ with arbitrarily prescribed singular sets. |
We consider eigenfunctions of a semiclassical Schr{ö}dinger operator on an interval, with a single-well type potential and Dirichlet boundary conditions. We give upper/lower bounds on the L^2 density of the eigenfunctions that are uniform in both semiclassical and high energy limits. |
For a homogeneous polynomial $p$ in $\xi\in {\bf R}^n$ with Gevrey coefficients, it is known that the Cauchy problem for any realization of $p$ is well-posed in the Gevrey class of order $s<2$ if the characteristic roots are real. In this note, we give examples showing the situation of the converse direction, in pa... |
This paper deals with $(K_1, K_2)$-quasiregular mappings. It is shown, by Morrey's Lemma and isoperimetric inequality, that every $(K_1, K_2)$-quasiregular mapping satisfies a Hölder condition with exponent $\alpha$ on compact subsets of its domain, where \begin{align} \alpha=\begin{cases} 1/K_1, & \text{for }... |
It is well recognized that in auxiliary equation methods, the exact solutions of different types of auxiliary equations may produce new types of exact travelling wave solutions to nonlinear partial differential equations in hand. In this study, we extend the class of auxiliary equations of classical Bernoulli equation... |
We consider the sinh-Poisson equation $$(P)_\lambda\quad -\Delta u=\la\sinh u\ \hbox{in}\ \Omega,\ u=0\ \hbox{on}\ \partial\Omega,$$ where $\Omega$ is a smooth bounded domain in $\rr^2$ and $\lambda$ is a small positive parameter. <br>If $0\in\Omega$ and $\Omega$ is symmetric with respect to the origin, for any intege... |
In this paper we present the solution to the problem of recovering rather arbitrary integral operator based on incomplete information with error. We apply the main result to obtain optimal methods of recovery and compute the optimal error for the solutions to certain integral equations as well as boundary and initial ... |
In this article, we show that the continuous data assimilation algorithm is valid for the 3D primitive equations of the ocean. Namely, the $L^2$ norm of the assimilated solution converge to that of the reference solution at an exponential rate in time. |
In this paper, we prove a maximum principle for the general multi-term space-time-fractional transport equation and apply it for establishing uniqueness of solution to an initial-boundary-value problem for this equation. We also derive some comparison principles for solutions to the initial-boundary-value problems wit... |
Let $(M,g)$ be a compact Riemannian surface without boundary, $W^{1,2}(M)$ be the usual Sobolev space, $J: W^{1,2}(M)\rightarrow \mathbb{R}$ be the functional defined by $$J(u)=\frac{1}{2}\int_M|\nabla u|^2dv_g+8\pi \int_M udv_g-8\pi\log\int_Mhe^udv_g,$$ where $h$ is a positive smooth function on $M$. In an inspiring ... |
In the exciton-polariton system, a linear dispersive photon field is coupled to a nonlinear exciton field. Short-time analysis of the lossless system shows that, when the photon field is excited, the time required for that field to exhibit nonlinear effects is longer than the time required for the nonlinear Schrödinge... |
In this paper we develop in detail the geometric constructions that lead to many uniqueness results for the determination of polyhedral sets, typically scatterers, by a finite minimal number of measurements. We highlight how unique continuation and a suitable reflection principle are enough to proceed with the constru... |
We show in this paper a Lipschitz stability result for a crack inverse problem in half space. The direct problem is a Laplace equation with zero Neumann condition on the top boundary. |
\noindent We are concerned with positive normalized solutions $(u,\lambda)\in H^1(\mathbb{R}^2)\times\mathbb{R}$ to the following semi-linear Schrödinger equations $$ -\Delta u+\lambda u=f(u), \quad\text{in}~\mathbb{R}^2, $$ satisfying the mass constraint $$\int_{\mathbb{R}^2}|u|^2\, dx=c^2\ . $$ We are interested in t... |
We consider planar solutions to certain quasilinear elliptic equations subject to the Dirichlet boundary conditions; the boundary data is assumed to have finite number of relative maximum and minimum values. We are interested in certain vanishing properties of sign changing solutions to such a Dirichlet problem. |
The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. |
Ranges of the real-valued parameters $\alpha$, $a$, $b$, and $m$ are identified for which the operator $$\mathcal{A}_{\alpha}(a,b)f(x):=x^\alpha\left(f''(x)+\frac{a}{x}f'(x)+\frac{b}{x^2}f(x)\right), \quad x>0,$$ generates an analytic semigroup in $L_1((0,\infty),x^m\mathrm{d}x)$. |
Based on recent well-posedness results in Sobolev (or Besov spaces) for periodic solutions to the Fornberg-Whitham equations we investigate here the questions of wave breaking and blow-up for these solutions. We show first that finite maximal life time of a solution necessarily leads to wave breaking. |
In this paper, we are concerned with the limiting absorption principle for the fractional Helmholtz equation, By establishing the boundedness estimate for the resolvent of fractional Helmholtz operator, we obtain the nontrivial Lq(Rn) complex valued solutions for (0.1). By setting up a dual variational framework, we a... |
In this paper, we deal with anisotropic singular perturbations of some class of elliptic problem. We study the asymptotic behavior of the solution in certain second order pseudo Sobolev space. |
We consider multivalued maps between $\Omega \subset \mathbb{R}^N$ open ($N \ge 2$) and a smooth, compact Riemannian manifold $\mathcal{N}$ locally minimizing the Dirichlet energy. An interior partial Hölder regularity result in the spirit of R. Schoen and K. Uhlenbeck is presented. |
One of the classical models in mathematical biology is the Lotka-Volterra competition model, describing the dynamics of two populations competing for resources. Two possible regimes in this system are given by their coexistence or extinction of a weaker population. |
Let L be a second order, uniformly elliptic operator, and consider the equation L u=f under the homogeneous boundary condition u=0. It is well known that f in C(Om) (Om= Omega) does not guarantee second order derivatives D^2 u in C(Om). |
The inverse source problem where an unknown source is to be identified from the knowledge of its radiated wave is studied. The focus is placed on the effect that multi-frequency data has on establishing uniqueness. |
We study the mean values sets of the second order divergence form elliptic operator with principal coefficients defined as $$a^{ij}_k(x):= \begin{cases} \alpha_k \delta^{ij}(x) &x_n>0 \beta_k \delta^{ij}(x) &x_n<0. \end{cases}$$ In particular, we will show that the mean value sets associated to such an o... |
We are concerned with Dirichlet problems of the form $${\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f (u)=0\ \mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial\Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1<p<n$ and $f$ is a continuous function with supercritical growth from the viewpo... |
The Kompaneets equation describes a field of photons exchanging energy by Compton scattering with the free electrons of a homogeneous, isotropic, non-relativistic, thermal plasma. This paper strives to advance our understanding of how this equation captures the phenomenon of Bose-Einstein condensation through the stud... |
The global existence of smooth solutions to the vacuum free boundary problem with physical singularity of compressible Euler equations with damping and gravity is proved in space dimensions $n=1, 2, 3$, for the initial data being small perturbations of the stationary solution. Moreover, the exponential decay of the ve... |
In this paper, we build a Gibbs measure for the 1d cubic Klein-Gordon equation on $\mathbb R$ with a decreasing non linearity, in the sense that the non linearity $f^3$ is multiplied by $\chi$ where $\chi$ is a sufficiently integrable non negative function. We prove that this equation is almost surely globally well-po... |
In this paper, we consider the phenomenon of monostable pulsating fronts for multi-dimensional reaction-diffusion-advection systems in periodic media. Recent results have addressed the existence of pulsating fronts and the linear determinacy of spreading speed (Du, Li and Shen, \textit{J. |
We consider a class of linear Schrödinger equations in R^d with rough Hamiltonian, namely with certain derivatives in the Sjöstrand class $M^{\infty,1}$. We prove that the corresponding propagator is bounded on modulation spaces. |
We prove locally in time the existence of a smooth solution for multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type. We establish also natural Hölder class for the boundary conditions in the Cauchy-Dirichlet problem for a degenerate parabolic equation. |
The authors show that bilinear estimates for null forms hold for Dirichlet-wave equations outside of convex obstacle. This generalizes results for the Euclidean case of Klainerman and Machedon, and of Sogge for the variable coefficient boundaryless case. |
With the help of some techniques based on certain inverse pairs of symbolic operators, the authors investigated several decomposition formulas associated with Srivastava's Hypergeometric functions of three variables. Some operator identities have been constructed in this matter. |
The paper is devoted to investigating the well-posedness, stability and large-time behavior near the hydrostatic balance for the 2D Boussinesq equations with partial dissipation. More precisely, the global well-posedness is obtained in the case of partial viscosity and without thermal diffusion for the initial data be... |
As a starting point of studying the long time behavior of the $3D$ water waves system in the flat bottom setting, in this paper, we try to improve the understanding of the Dirichlet-Neumann operator in this setting. As an application, we study the $3D$ gravity waves system and derive a new $L^2-L^\infty$ type energy e... |
We discuss a variational approach to doubly nonlinear wave equations of the form $\rho u_{tt} + g (u_t) - \Delta u + f (u)=0$. This approach hinges on the minimization of a parameter-dependent family of uniformly convex functionals over entire trajectories, namely the so-called Weighted Inertia-Dissipation-Energy (WID... |
It is known that the Kuramoto-Velarde equation is globally well-posed on Sobolev spaces in the case when the parameters $\gamma_1$ and $\gamma_2$ involved in the non-linear terms verify $ \gamma_1=\frac{\gamma_1}{2}$ or $\gamma_2=0$. In the complementary case of these parameters, the global existence or blow-up of sol... |
We consider a coupled system of Schrödinger equations, arising in quantum mechanics via the so-called time-dependent self-consistent field method. Using Wigner transformation techniques we study the corresponding classical limit dynamics in two cases. |
Localized energy estimates have become a fundamental tool when studying wave equations in the presence of asymptotically at background geometry. Trapped rays necessitate a loss when compared to the estimate on Minkowski space. |
In this paper we prove that the motion of a solid body in a two dimensional incompressible perfect fluid converges, when the body shrinks to a point with fixed mass and circulation, to a variant of the vortex-wave system where the vortex, placed in the point occupied by the shrunk body, is accelerated by a lift force s... |
We show that if $u$ solves the fractional parabolic equation $(\partial_t - \Delta )^s u = Vu$ in $B_5 \times (-25, 0]$ ($0<s<1$) such that $u(\cdot, 0) \not\equiv 0$, then the maximal vanishing order of $u$ in space-time at $(0,0)$ is upper bounded by $C\left(1+\|V\|_{C^{1}_{(x,t)}}^{1/2s}\right)$. As $s \to 1$... |
We propose a suitable analytical framework to perform numerical analysis of problems arising in compressible fluid models with uncertain data. We discuss both weak and strong stochastic approach, where the former is based on the knowledge of the mere distribution (law) of the random data typical for the Monte-Carlo an... |
We consider the Kudryashov-Sinelshchikov equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation coverge to the entropy ones of the Burgers equation. |
A Hörmander-type theorem is established for Itô processes and related backward stochastic partial differential equations (BSPDEs). A short self-contained proof is also provided for the $L^2$-theory of linear, possibly degenerate BSPDEs, in which new gradient estimates are obtained. |
In this paper we present a characterization of the symmetric rotational periodic gravity water waves of finite depth and without stagnation points in terms of the underlying flow. Namely, we show that such a wave is symmetric and has a single crest and trough per period if and only if there exists a vertical line with... |
This Note is concerned with the asymptotic behavior of the minimal KPP speed of propagation for reaction- advection-diffusion equations with a large drift Mq (where q is the advection). We first give the limit of the speed as M\rightarrow+\infty in any space dimension N. Then, we give the necessary and sufficient cond... |
In this paper, we study the existence of positive functions $K \in C^1(\mathbb{S}^n)$ such that the conformal $Q$-curvature equation \begin{equation}\label{001} P_m (v) =K v^{\frac{n+2m}{n-2m}}~~~~~~ {on} ~ \mathbb{S}^n \{equation} has a singular positive solution $v$ whose singular set is a single point, where $m$ is ... |
This article is devoted to a generalized version of Smoluchowski's coagulation equation. This model describes the time evolution of a system of aggregating particles under the effect of external input and output particles. |
We give asymptotics for the level set equation for mean curvature flow on a convex domain near the point where it attains a maximum. It is known that solutions are not necessarily $C^3,$ and we recover this result and construct non-smooth solutions which are $C^3.$ We also construct solutions having prescribed behavio... |
We construct non-trivial weak solutions $\theta\in C_t^0C_x^{0-}$ to the surface quasi-geostrophic (SQG) equations, which have compact support in time and, thus, violate the conservation of the Hamiltonian. The result is sharp in view of the fact that such a conservation law holds for all weak solutions in the class $... |
We present a waveform relaxation version of the Dirichlet-Neumann method for parabolic problem. Like the Dirichlet-Neumann method for steady problems, the method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves with Dirichlet boundary conditions followed by subdom... |
We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterise all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on a given interval, and then apply this result... |
In this paper, we investigate the existence of nonnegative solutions for the problem $$ -\mathcal{L}_{K}u+V(x)u=f(u) $$ in $\mathbb R^n$, where $-\mathcal{L}_{K}$ is a integro-differential operator with measurable kernel $K$ and $V$ is a continuous potential. Under apropriate hypothesis, we prove, using variational me... |
In this work, we mainly focus on the energy-supercritical nonlinear Schrödinger equation, $$ <br>i\partial_{t}u+\Delta u= \mu|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $\mu=\pm1$ and $p>\frac4{d-2}$. %In this work, we consider the energy-supercritical cases, that is, $p\in (\frac4{d-2},+\infty)$. |
We construct approximate solutions $ (\psi_*, n_*)$ of the critical 4D Zakharov system which collapse in finite time to a singular renormalization of the solitary bulk solutions $ (\lambda e^{i \theta}W, \lambda^2 W^2)$ . To be precise for $ N \in \mathbb{Z}_+,\;N \gg1 $ we obtain a magnetic envelope/ion density pair ... |
In this paper, we determine the convergence speed of an upscaling of a pseudo-parabolic system containing drift terms with scale separation of size $\epsilon \ll 1$. Both the upscaling and convergence speed determination exploit a natural spatial-temporal decomposition, which splits the pseudo-parabolic system into a ... |
We address semigroup well-posedness for a linear, compressible viscous fluid interacting at its boundary with an elastic plate. We derive the model by linearizing the compressible Navier-Stokes equations about an arbitrary flow state, so the fluid PDE includes an ambient flow profile $\mathbf{U}$. |
We investigate into the regularity of $\mathrm{BV}^{\mathbb{A}}$-minimisers for $\mathbb{C}$-elliptic differential operators $\mathbb{A}$ in $2$ dimensions. Our studies strongly rely on the special structure of such differential operators. |
This paper considers a class of nonlinear time harmonic Maxwell systems at fixed frequency, with nonlinear terms taking the form $\mathscr{X}(x,|\vec E(x)|^2)\vec E(x)$, $\mathscr{Y}(x,|\vec H(x)|^2)\vec H(x)$, such that $\mathscr{X}(x,s)$, $\mathscr{Y}(x,s)$ are both real analytic in $s$. Such nonlinear terms appear ... |
Let $\mathcal B$ be a sufficiently smooth rigid body (compact set of $\mathbb R^3$) of arbitrary shape moving in an unbounded Navier-Stokes liquid under the action of prescribed external force, $\textup{F}$, and torque, $\textup{M}$. We show that if the data are suitably regular and small, and $\textup{F}$ and $\textu... |
In this note we devise and analyze a well-posed variational formulation of the Neumann boundary value problem associated to the biharmonic operator $\Delta^2$. An alternative formulation as a system of two Poisson problems for the Laplace operator $\Delta$ is also derived. |
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