Analysis of PDEs (math.AP)

Abstract: We solve the Schr{\"o}dinger equation with logarithmic nonlinearity and multiplicative spatial white noise on R d with d $\le$ 2. Because of the nonlinearity, the regularity structures and the paracontrolled calculus can not be used. To solve the equation, we rely on an exponential transform that has proven useful in the context of other singular SPDEs.

Abstract: We prove new optimal $C^{1,\alpha}$ regularity results for obstacle problems involving evolutionary $p$-Laplace type operators in the degenerate regime $p > 2$. Our main results include the optimal regularity improvement at free boundary points in intrinsic backward $p$-paraboloids, up to the critical exponent, $\alpha \leq 2/(p-2)$, and the optimal regularity across the free boundaries in the full cylinders up to a universal threshold. Moreover, we provide an intrinsic criterion by which the optimal regularity improvement at free boundaries can be extended to the entire cylinders. An important feature of our analysis is that we do not impose any assumption on the time derivative of the obstacle. Our results are formulated in function spaces associated to what we refer to as higher order or $C^{1,\alpha}$ intrinsic interpolative geometries.

Abstract: We consider the Hartree and Schr\"{o}dinger equations describing the time evolution of wave functions of infinitely many interacting fermions in three-dimensional space. These equations can be formulated using density operators, and they have infinitely many stationary solutions. In this paper, we prove the asymptotic stability of a wide class of stationary solutions. We emphasize that our result includes Fermi gas at zero temperature. This is one of the most important steady states from the physics point of view; however, its asymptotic stability has been left open after Lewin and Sabin first formulated this stability problem and gave significant results in their seminal work [arXiv:1310.0604].

Abstract: Let $\Omega \subset \mathbb{R}^d$ and consider the magnetic Laplace operator given by $ H(A) = \left(- i\nabla - A(x)\right)^2$, where $A:\Omega \rightarrow \mathbb{R}^d$, subject to Dirichlet eigenfunction. This operator can, for certain vector fields $A$, have eigenfunctions $H(A) \psi = \lambda \psi$ that are highly localized in a small region of $\Omega$. The main goal of this paper is to show that if $|\psi|$ assumes its maximum in $x_0 \in \Omega$, then $A$ behaves `almost' like a conservative vector field in a $1/\sqrt{\lambda}-$neighborhood of $x_0$ in a precise sense: we expect localization in regions where $\left|\mbox{curl} A \right|$ is small. The result is illustrated with numerical examples.

Abstract: For a function $f$ which foliates a one-sided neighbourhood of a closed hypersurface $M$, we give an estimate of the distance of $M$ to a Wulff shape in terms of the $L^{p}$-norm of the traceless $F$-Hessian of $f$, where $F$ is the support function of the Wulff shape. This theorem is applied to prove quantitative stability results for the anisotropic Heintze-Karcher inequality, the anisotropic Alexandrov problem, as well as for the anisotropic overdetermined boundary value problem of Serrin-type.

Abstract: This paper concerns an age-structured juvenile-adult model with harvesting pulse and moving boundaries in a heterogeneous environment, in which the moving boundaries describe the natural expanding front of species and human periodic pulse intervention is carried on the adults. The principal eigenvalue is firstly defined and its properties involving the intensity of harvesting and length of habitat sizes are analysed. Then the criteria to determine whether the species spread or vanish is discussed, and some relevant sufficient conditions characterized by pulse are established. Our results reveal that the co-extinction or coexistence of species is influenced by internal expanding capacities from species itself and external harvesting pulse from human intervention, in which the intensity and timing of harvesting play key roles. The final numerical approximations indicate that the larger the harvesting rate and the shorter the harvesting period, the worse the survival of all individuals due to the cooperation among juveniles and adults, and such harvesting pulse can even alter the situation of species, from persistence to extinction. In addition, expanding capacities also affect or alter the outcomes of spreading-vanishing

Abstract: We construct and analyze solutions to a regularized homogeneous $p$-harmonic map flow equation for general $p \geq 2$. The homogeneous version of the problem is new and features a monotonicity formula extending the one found by Struwe for $p = 2$; such a formula is not available for the nonhomogeneous equation. The construction itself is via a Ginzburg-Landau-type approximation \`a la Chen-Struwe, employing tools such as a Bochner-type formula and an $\varepsilon$-regularity theorem. We similarly obtain strong subsequential convergence of the approximations away from a concentration set with parabolic codimension at least $p$. However, the quasilinear and non-divergence nature of the equation presents new obstacles that do not appear in the classical case $p = 2$, namely uniform-time existence for the approximating problem, and thus our basic existence result is stated conditionally.

Abstract: The parabolic integro-differential Cauchy problem with spatially dependent coefficients is considered in generalized Bessel potential spaces where smoothness is defined by L\'evy measures with O-regularly varying profile. The coefficients are assumed to be bounded and H\"older continuous in the spatial variable. Our results can cover interesting classes of L\'evy measures that go beyond those comparable to $dy/\left|y\right|^{d+\alpha}$.

Abstract: In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -\Delta_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $\Omega$,} \newline u\geq 0 & \text{in $\Omega$,} \newline u=0 & \text{on $\partial \Omega$,} \ \end{cases} \end{equation*} in a domain $\Omega \subset \mathbb{R}^{N}$ $(N \geq 2)$, where $1\leq p<N $, $g$ is a positive and continuous function on $[0,\infty)$, and $h$ is a continuous function on $[0,\infty)$ (possibly blowing up at the origin). We show how the presence of regularizing terms $h$ and $g$ allows to prove existence of finite energy solutions for nonnegative data $f$ only belonging to $L^1(\Omega)$.

Abstract: In this manuscript we construct global in time classical solutions to mean field games master equations in the lack of idiosyncratic noise in the individual agents' dynamics. These include both deterministic models and dynamics driven solely by a Brownian common noise. We consider a general class of non-separable Hamiltonians and final data functions that are supposed to be displacement monotone. Our main results unify and generalize in particular some of the well-posedness results on displacement monotone master equations obtained recently by Gangbo--M\'esz\'aros and Gangbo--M\'esz\'aros--Mou--Zhang.

Abstract: This article concerns the Cauchy problem for the gravity-capillary water waves system in general dimensions. We establish local well-posedness for initial data in $H^s$, with $s > \frac{d}{2} + 2 - \mu$, with $\mu = \frac{3}{14}$ and $\mu = \frac37$ in the cases $d = 1$ and $d \geq 2$ respectively. This represents an improvement over the the state-of-the-art low regularity theory in $d \geq 2$ dimensions.