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Register a new userGeneration results for vector-valued elliptic operators with unbounded coefficients in L^p spaces
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We consider a class of vector-valued elliptic operators with unbounded coefficients, coupled up to the first-order, in the Lebesgue space L^p(R^d;R^m) with p in (1,infty). Sufficient conditions to prove generation results of an analytic C_0-semigroup T(t), together with a characterization of the domain of its generator, are given. Some results related to the hypercontractivity and the ultraboundedness of the semigroup are also established.
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We study existence and uniqueness of the invariant measure for a stochastic process with degenerate diffusion, whose infinitesimal generator is a linear subelliptic operator in the whole space R N with coefficients that may be unbounded. Such a measure together with a Liouville-type theorem will play a crucial role in two applications: the ergodic problem studied through stationary problems with vanishing discount and the long time behavior of the solution to a parabolic Cauchy problem. In both cases, the constants will be characterized in terms of the invariant measure.
We consider a class of non-trivial perturbations ${mathscr A}$ of the degenerate Ornstein-Uhlenbeck operator in ${mathbb R}^N$. In fact we perturb both the diffusion and the drift part of the operator (say $Q$ and $B$) allowing the diffusion part to be unbounded in ${mathbb R}^N$. Assuming that the kernel of the matrix $Q(x)$ is invariant with respect to $xin {mathbb R}^N$ and the Kalman rank condition is satisfied at any $xin{mathbb R}^N$ by the same $m<N$, and developing a revised version of Bernsteins method we prove that we can associate a semigroup ${T(t)}$ of bounded operators (in the space of bounded and continuous functions) with the operator ${mathscr A}$. Moreover, we provide several uniform estimates for the spatial derivatives of the semigroup ${T(t)}$ both in isotropic and anisotropic spaces of (Holder-) continuous functions. Finally, we prove Schauder estimates for some elliptic and parabolic problems associated with the operator ${mathscr A}$.
We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assumed to be only measurable either in the time or the spatial variable. As functions of the other variables the coefficients have small bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are allowed to blow up near the boundary with a certain optimal growth condition. As a corollary, we also obtain the corresponding results for elliptic equations.
We consider the Schrodinger operator on $[0,1]$ with potential in $L^1$. We prove that two potentials already known on $[a,1]$ ($ain(0,{1/2}]$) and having their difference in $L^p$ are equal if the number of their common eigenvalues is sufficiently large. The result here is to write down explicitly this number in terms of $p$ (and $a$) showing the role of $p$.
We consider wave equations with time-independent coefficients that have $C^{1,1}$ regularity in space. We show that, for nontrivial ranges of $p$ and $s$, the standard inhomogeneous initial value problem for the wave equation is well posed in Sobolev spaces $mathcal{H}^{s,p}_{FIO}(mathbb{R}^{n})$ over the Hardy spaces $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$ for Fourier integral operators introduced recently by the authors and Portal, following work of Smith. In spatial dimensions $n = 2$ and $n=3$, this includes the full range $1 < p < infty$. As a corollary, we obtain the optimal fixed-time $L^{p}$ regularity for such equations, generalizing work of Seeger, Sogge and Stein in the case of smooth coefficients.
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