Narrow Results

Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we investigate compactness for fourth order critical equations like P_gu = u^2♯-1, where is a Paneitz–Branson operator with constant coefficients b and c, u is required to be positive, and is critical from the Sobolev viewpoint. We prove that such equations are compact on locally conformally flat manifolds, unless b lies in some closed interval associated to the spectrum of the smooth symmetric (2,0)-tensor field involved in the definition of the geometric Paneitz–Branson operator.

We study the Dirichlet boundary value problem on a bounded domain Ω ⊂ ℝ^N. For 2 ≤ N ≤ 7, we characterize compactness for solutions sequence in terms of spectral informations. As a by-product, we give an uniqueness result for λ close to 0 and λ* in the class of all solutions with finite Morse index, λ* being the extremal value associated to the nonlinear eigenvalue problem.

In this paper, we obtain gradient continuity estimates for viscosity solutions of ${\mathrm{\Delta }}_p^{N}u=f$ in terms of the scaling critical $L(n,1)$-norm of $f$, where ${\mathrm{\Delta }}_p^N$ is the normalized $p$-Laplacian operator. Our main result corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potential ${\widetilde{𝕀}}_q^f$. Moreover, for $f\in L^m$ with $m>n$, we also obtain $C^{1,\alpha }$ estimates. This improves one of the regularity results in [A. Attouchi, M. Parviainen and E. Ruosteenoja, $C^{1,\alpha }$ regularity for the normalized $p$-Poisson problem, J. Math. Pures Appl. (9)  108(4) (2017) 553–591], where a $C^{1,\alpha }$ estimate was established depending on the $L^m$-norm of $f$ under the additional restriction that $p>2$ and $m>\mathrm{\text{max}}(2,n,\frac{p}{2})$. We also mention that differently from the approach in the above paper, which uses methods from divergence form theory and nonlinear potential theory, the method in this paper is more non-variational in nature, and it is based on separation of phases inspired by the ideas in [L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations 107(2) (1994) 341–350]. Moreover, for $f$ continuous, our approach also gives a somewhat different proof of the $C^{1,\alpha }$ regularity result.

We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is $n\ge 7$ and the trace-free part of the second fundamental form is nonzero everywhere on the boundary.

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 $(l,\beta )$.

In this paper, we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in [B. Kirchheim, S. Müller and V. Šverák, Studying Nonlinear PDE by Geometry in Matrix Space (Springer, 2003), Sec. 7], and many of its properties have already been shown in [A. Lorent and G. Peng, Null Lagrangian measures in subspaces, compensated compactness and conservation laws, Arch. Ration. Mech. Anal. 234(2) (2019) 857–910; A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156]. In particular, in [A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156], it is shown that the differential inclusion does not contain any $T_4$ configurations. Here, we continue that study by showing that the differential inclusion does not contain $T_5$ configurations.