Search results for "FOS: Mathematics"
showing 10 items of 1448 documents
A minimization problem with free boundary and its application to inverse scattering problems
2023
We study a minimization problem with free boundary, resulting in hybrid quadrature domains for the Helmholtz equation, as well as some application to inverse scattering problem.
Some sufficient conditions for lower semicontinuity in SBD and applications to minimum problems of Fracture Mechanics
2009
We provide some lower semicontinuity results in the space of special functions of bounded deformation for energies of the type $$ %\int_\O {1/2}({\mathbb C} \E u, \E u)dx + \int_{J_{u}} \Theta(u^+, u^-, \nu_{u})d \H^{N-1} \enspace, \enspace [u]\cdot \nu_u \geq 0 \enspace {\cal H}^{N-1}-\hbox{a. e. on}J_u, $$ and give some examples and applications to minimum problems. \noindent Keywords: Lower semicontinuity, fracture, special functions of bounded deformation, joint convexity, $BV$-ellipticity.
A thermoelastic theory with microtemperatures of type III
2021
In this paper, we use the Green-Naghdi theory of thermomechanics of continua to derive a nonlinear theory of thermoelasticity with microtemperatures of type III. This theory permits propagation of both thermal and microtemperatures waves at finite speeds with dissipation of energy. The equations of the linear theory are also obtained. With the help of the semigroup theory of linear operators we establish that the linear anisotropic problem is well posed and we study the asymptotic behavior of the solutions. Finally, we investigate the impossibility of the localization in time of solutions.
Asymptotic behavior for the heat equation in nonhomogeneous media with critical density
2012
We study the asymptotic behavior of solutions to the heat equation in nonhomogeneous media with critical singular density $$ |x|^{-2}\partial_{t}u=\Delta u, \quad \hbox{in} \ \real^N\times(0,\infty). $$ The asymptotic behavior proves to have some interesting and quite striking properties. We show that there are two completely different asymptotic profiles depending on whether the initial data $u_0$ vanishes at $x=0$ or not. Moreover, in the former the results are true only for radially symmetric solutions, and we provide counterexamples to convergence to symmetric profiles in the general case.
Volume growth, capacity estimates, p-parabolicity and sharp integrability properties of p-harmonic Green functions
2023
In a complete metric space equipped with a doubling measure supporting a $p$-Poincar\'e inequality, we prove sharp growth and integrability results for $p$-harmonic Green functions and their minimal $p$-weak upper gradients. We show that these properties are determined by the growth of the underlying measure near the singularity. Corresponding results are obtained also for more general $p$-harmonic functions with poles, as well as for singular solutions of elliptic differential equations in divergence form on weighted $\mathbf{R}^n$ and on manifolds. The proofs are based on a new general capacity estimate for annuli, which implies precise pointwise estimates for $p$-harmonic Green functions…
Variational principles for fluid dynamics on rough paths
2022
In this paper, we introduce a new framework for parametrization schemes (PS) in GFD. Using the theory of controlled rough paths, we derive a class of rough geophysical fluid dynamics (RGFD) models as critical points of rough action functionals. These RGFD models characterize Lagrangian trajectories in fluid dynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps. Three constrained variational approaches are formulated for the derivation of these models. The first is the Clebsch formulation, in which the constraints are imposed as rough advection laws. The second is the Hamilton-Pontryagin formulation, in which the constraints are imposed as right-invariant rough vector…
Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density
2013
We study the large time behavior of solutions to the porous medium equation in nonhomogeneous media with critical singular density $$ |x|^{-2}\partial_{t}u=��u^m, \quad \hbox{in} \ \real^N\times(0,\infty), $$ where $m>1$ and $N\geq3$. The asymptotic behavior proves to have some interesting and striking properties. We show that there are different asymptotic profiles for the solutions, depending on whether the continuous initial data $u_0$ vanishes at $x=0$ or not. Moreover, when $u_0(0)=0$, we show the convergence towards a profile presenting a discontinuity in form of a shockwave, coming from an unexpected asymptotic simplification to a conservation law, while when $u_0(0)>0$, the li…
On the modified fractional Korteweg-de Vries and related equations
2020
We consider in this paper modified fractional Korteweg-de Vries and related equations (modified Burgers-Hilbert and Whitham). They have the advantage with respect to the usual fractional KdV equation to have a defocusing case with a different dynamics. We will distinguish the weakly dispersive case where the phase velocity is unbounded for low frequencies and tends to zero at infinity and the strongly dispersive case where the phase velocity vanishes at the origin and goes to infinity at infinity. In the former case, the nonlinear hyperbolic effects dominate for large data, leading to the possibility of shock formation though the dispersive effects manifest for small initial data where scat…
Quantitative uniqueness estimates for $p$-Laplace type equations in the plane
2016
In this article our main concern is to prove the quantitative unique estimates for the $p$-Laplace equation, $1\max\{p,2\}$ or $q=p>2$, if $\|u\|_{L^\infty(\mathbb{R}^2)}\leq C_0$, then $u$ satisfies the following asymptotic estimates at $R\gg 1$ \[ \inf_{|z_0|=R}\sup_{|z-z_0|<1} |u(z)| \geq e^{-CR^{1-\frac{2}{q}}\log R}, \] where $C$ depends only on $p$, $q$, $\tilde{M}$ and $C_0$. When $q=\max\{p,2\}$ and $p\in (1,2]$, under similar assumptions, we have \[ \inf_{|z_0|=R} \sup_{|z-z_0|<1} |u(z)| \geq R^{-C}, \] where $C$ depends only on $p$, $\tilde{M}$ and $C_0$. As an immediate consequence, we obtain the strong unique continuation principle (SUCP) for nontrivial solutions of this equatio…
Kernel estimates for nonautonomous Kolmogorov equations with potential term
2014
Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients and a possibly unbounded potential term.