Narrow Results

We study a homogeneous infinite dimensional Dirichlet problem in a half-space of a Hilbert space involving a second-order elliptic operator with Hölder continuous coefficients. Thanks to a new explicit formula for the solution in the constant coefficients case, we prove an optimal regularity result of Schauder type. The proof uses nonstandard techniques from semigroups and interpolation theory and involves extensive computations on Gaussian integrals.

For a conformally compact Poincaré–Einstein manifold $(X,g_{+})$, we consider two types of compactifications for it. One is $ḡ={\rho }^2{g}_{+}$, where $\rho$ is a fixed smooth defining function; the other is the adapted (including Fefferman–Graham) compactification ${ḡ}_s={\rho }_s^2{g}_{+}$ with a continuous parameter $s>\frac{n}{2}$. In this paper, we mainly prove that for a set of conformally compact Poincaré–Einstein manifolds $\{(X,g_{+}^{(i)})\}$ with conformal infinity of positive Yamabe type, $\{{ḡ}^{(i)}\}$ is compact in $C^{k,\alpha }(\overline{X})$ topology if and only if $\{{ḡ}_s^{(i)}\}$ is compact in some $C^{l,\beta }(\overline{X})$ topology, provided that ${ḡ}^{(i)}{|}_{TM}={ḡ}_s^{(i)}{|}_{TM}={ĝ}^{(i)}$ and ${ĝ}^{(i)}$ has positive scalar curvature for each $i$. See Theorem 1.1 and Corollary 1.1 for the exact relation of $(k,\alpha )$ and $(l,\beta )$.

We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm $H^{2+}$ of the variables in the “wave-part” and the Hölder norm $C^{0,0+}$ of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption $H^{2+}$ is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory.