Analysis of PDEs (math.AP)

Abstract: We study linear dispersive equations in dimension one and two for a class of radial nonhomogeneous phases. L 1 $\rightarrow$ L $\infty$ type estimates, Strichartz estimates, local Kato smoothing and Morawetz type estimates are provided. We then apply our results to different water wave models.

Abstract: In this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic-elliptic Keller-Segel system in the framework of whole spaces detailized by Euclid space $\mathbb{R}^n\,\,(n \geqslant 4)$ and hyperbolic space $\mathbb{H}^n\,\, (n \geqslant 2)$. Our method is based on the dispersive and smoothing estimates of the heat semiroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviour of periodic mild solutions of Keller-Segel system obtained in $\mathbb{R}^n$ and the one in $\mathbb{H}^n$.

Abstract: In this work we study an inverse problem for the minimal surface equation on a Riemannian manifold $(\mathbb{R}^{n},g)$ where the metric is of the form $g(x)=c(x)(\hat{g}\oplus e)$. Here $\hat{g}$ is a simple Riemannian metric on $\mathbb{R}^{n-1}$, $e$ is the Euclidean metric on $\mathbb{R}$ and $c$ a smooth positive function. We show that if we know the associated Dirichlet-to-Neumann maps corresponding to metrics $g$ and $\tilde{c}g$, then the Taylor series of the conformal factor $\tilde{c}$ at $x_n=0$ is equal to a positive constant. We also show a partial data result when $n=3$.

Abstract: The exponential decay rate of the semigroup $S(t)=e^{t\mathbb{A}}$ generated by the abstract damped wave equation $$\ddot u + 2f(A) \dot u +A u=0 $$ is here addressed, where $A$ is a strictly positive operator. The continuous function $f$, defined on the spectrum of $A$, is subject to the constraints $$\inf f(s)>0\qquad\text{and}\qquad \sup f(s)/s <\infty$$ which are known to be necessary and sufficient for exponential stability to occur. We prove that the operator norm of the semigroup fulfills the estimate $$\|S(t)\|\leq Ce^{\sigma_*t}$$ being $\sigma_*<0$ the supremum of the real part of the spectrum of $\mathbb{A}$. This estimate always holds except in the resonant cases, where the negative exponential $e^{\sigma_*t}$ turns out to be penalized by a factor $(1+t)$. The decay rate is the best possible allowed by the theory.

Abstract: In this paper we prove some integral estimates on the minimal growth of the positive part $u_+$ of subsolutions of quasilinear equations \[ \mathrm{div} A(x,u,\nabla u) = V|u|^{p-2}u \] on complete Riemannian manifolds $M$, in the non-trivial case $u_+\not\equiv 0$. Here $A$ satisfies the structural assumption $|A(x,u,\nabla u)|^{p/(p-1)} \leq k \langle A(x,u,\nabla u),\nabla u\rangle$ for some constant $k>0$ and for $p>1$ the same exponent appearing on the RHS of the equation, and $V$ is a continuous positive function, possibly decaying at a controlled rate at infinity. We underline that the equation may be degenerate and that our arguments do not require any geometric assumption on $M$ beyond completeness of the metric. From these results we also deduce a Liouville-type theorem for sufficiently slowly growing solutions.

Abstract: We establish a regularity result for optimal sets of the isoperimetric problem with double density under mild H\"older regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach the optimal regularity class $C^{1,\frac{\alpha}{2-\alpha}}$ in any dimension.

Abstract: We consider solutions satisfying the Neumann zero boundary condition and a linearized mean field game system in $\Omega \times (0,T)$, where $\Omega$ is a bounded domain in $\mathbb{R}^d$ and $(0,T)$ is the time interval. We prove two kinds of stability results in determining the solutions. The first is H\"older stability in time interval $(\epsilon, T)$ with arbitrarily fixed $\epsilon>0$ by data of solutions in $\Omega \times \{T\}$. The second is the Lipschitz stability in $\Omega \times (\epsilon, T-\epsilon)$ by data of solutions in arbitrarily given subdomain of $\Omega$ over $(0,T)$.

Abstract: A standard elasto-plasto-dynamic model at finite strains based on the Lie-Liu-Kr\"oner multiplicative decomposition, formulated in rates, is here enhanced to cope with spatially inhomogeneous materials by using the reference (called also return) mapping. Also an isotropic hardening can be involved. Consistent thermodynamics is formulated, allowing for both the free and the dissipation energies temperature dependent. The model complies with the energy balance and entropy inequality. A multipolar Stokes-like viscosity and plastic rate gradient are used to allow for a rigorous analysis towards existence of weak solutions by a semi-Galerkin approximation.

Abstract: We consider the harmonic map heat flow for maps from the plane to the two-sphere. It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a subsequence along which bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubble configurations in continuous time.