We consider a wide class of functionals with the property of changing their growth and ellipticity properties according to the modulating coefficients in the framework of Musielak–Orlicz spaces. In particular, we provide an optimal condition on the modulating coefficient to establish the Hölder regularity and Harnack inequality for quasiminimizers of the generalized double phase functional with $\left(G,H\right)$-growth for two Young functions $G$ and $H$.

We prove an $𝜖$-regularity theorem for vector-valued $p$-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere.

This does not seem to follow from the reflection method that Scheven used for harmonic maps with free boundary (i.e., the case $p=2$): the reflected equation can be interpreted as a $p$-harmonic map equation into a manifold, but the regularity theory for such equations is only known for round targets.

Instead, we follow the spirit of Schikorra’s recent work on free boundary harmonic maps and choose a good frame directly at the free boundary. This leads to growth estimates, which, in the critical regime $p=n$, imply Hölder regularity of solutions. In the supercritical regime, $p<n$, we combine the growth estimate with the geometric reflection argument: the reflected equation is supercritical, but, under the assumption of growth estimates, solutions are regular.

In the case $p<n$, for stationary $p$-harmonic maps with free boundary, as a consequence of a monotonicity formula we obtain partial regularity up to the boundary away from a set of $\left(n-p\right)$-dimensional Hausdorff measure.

We prove uniform Sobolev estimates for the resolvent of Schrödinger operators with large scaling-critical potentials without any repulsive condition. As applications, global-in-time Strichartz estimates including some nonadmissible retarded estimates, a Hörmander-type spectral multiplier theorem, and Keller-type eigenvalue bounds with complex-valued potentials are also obtained.

We are interested in several questions raised mainly by Mancini and Martinazzi (2017) (see also work of McLeod and Peletier (1989) and Pruss (1996)). We consider the perturbed Moser–Trudinger inequality $I_{\alpha }^g\left(\mathrm{\Omega }\right)$ at the critical level $\alpha =4\pi$, where $g$, satisfying $g\left(t\right)\to 0$ as $t\to +\infty$, can be seen as a perturbation with respect to the original case $g\equiv 0$. Under some additional assumptions, ensuring basically that $g$ does not oscillate too fast as $t\to +\infty$, we identify a new condition on $g$ for this inequality to have an extremal. This condition covers the case $g\equiv 0$ studied by Carleson and Chang (1986), Struwe (1988), and Flucher (1992). We prove also that this condition is sharp in the sense that, if it is not satisfied, $I_{4\pi }^g\left(\mathrm{\Omega }\right)$ may have no extremal.

We address the local well-posedness of the hydrostatic Navier–Stokes equations. These equations, sometimes called reduced Navier–Stokes/Prandtl, appear as a formal limit of the Navier–Stokes system in thin domains, under certain constraints on the aspect ratio and the Reynolds number. It is known that without any structural assumption on the initial data, real-analyticity is both necessary and sufficient for the local well-posedness of the system. In this paper we prove that for convex initial data, local well-posedness holds under simple Gevrey regularity.

We prove variation-norm estimates for certain oscillatory integrals related to Carleson’s theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schrödinger-like equations due to Lee, Rogers and Seeger. In dimension 1, our proof additionally relies on a local smoothing estimate. Though the known endpoint local smoothing estimate by Rogers and Seeger is more than sufficient for our purpose, we also give a proof of certain local smoothing estimates using Bourgain–Guth iteration and the Bourgain–Demeter ${\ell }^2$ decoupling theorem. This may be of independent interest, because it improves the previously known range of exponents for spatial dimensions $n\ge 4$.

Federer’s characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension 1. In complete metric spaces that are equipped with a doubling measure and support a Poincaré inequality, the “only if” direction was shown by Ambrosio (2002). By applying fine potential theory in the case $p=1$, we prove that the “if” direction holds as well.

We extend the results of our paper “Attractors for two-dimensional waves with homogeneous Hamiltonians of degree 0,” written with Laure Saint-Raymond, to the case of forced linear wave equations in any dimension. We prove that, in dimension 2, if the foliation on the boundary at infinity of the energy shell is Morse–Smale, we can apply Mourre’s theory and hence get the asymptotics of the forced solution. We also characterize the wavefront sets of the limit Schwartz distribution using radial propagation estimates.

We show that the derivative nonlinear Schrödinger (DNLS) equation is globally well-posed in the weighted Sobolev space $H^{2,2}\left(ℝ\right)$. Our result exploits the complete integrability of the DNLS equation and removes certain spectral conditions on the initial data required by our previous work, thanks to Zhou’s analysis (Comm. Pure Appl. Math. 42:7 (1989), 895–938) on spectral singularities in the context of inverse scattering.

We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $m\ge 3$ starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With the same methods, we show rigidity of hyperbolic space under the Yamabe flow.

Motivated by the inequality $\parallel f+g{\parallel }_2^2\le \parallel f{\parallel }_2^2+2\parallel fg{\parallel }_1+\parallel g{\parallel }_2^2$, Carbery (2009) raised the question of what is the “right” analogue of this estimate in $L^p$ for $p\ne 2$. Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an $L^p$ version of this inequality by providing upper bounds for $\parallel f+g{\parallel }_p^p$ in terms of the quantities $\parallel f{\parallel }_p^p$, $\parallel g{\parallel }_p^p$ and $\parallel fg{\parallel }_{p∕2}^{p∕2}$ when $p\in \left(0,1\right]\cup \left[2,\infty \right)$, and lower bounds when $p\in \left(-\infty ,0\right)\cup \left(1,2\right)$, thereby proving (and improving) the suggested possible inequalities of Carbery. We continue investigation in this direction by refining the estimates of Carlen, Frank, Ivanisvili and Lieb. We obtain upper bounds for $\parallel f+g{\parallel }_p^p$ also when $p\in \left(-\infty ,0\right)\cup \left(1,2\right)$ and lower bounds when $p\in \left(0,1\right]\cup \left[2,\infty \right)$. For $p\in \left[1,2\right]$ we extend our upper bounds to any finite number of functions. In addition, we show that all our upper and lower bounds of $\parallel f+g{\parallel }_p^p$ for $p\in ℝ$, $p\ne 0$, are the best possible in terms of the quantities $\parallel f{\parallel }_p^p$, $\parallel g{\parallel }_p^p$ and $\parallel fg{\parallel }_{p∕2}^{p∕2}$, and we characterize the equality cases.