Analysis & PDE最新文献

We study nodal sets of Steklov eigenfunctions in a bounded domain with ${\mathcal{𝒞}}^2$ boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that, for $u_{\lambda }$ a Steklov eigenfunction with eigenvalue $\lambda \ne 0$, we have ${\mathcal{\mathscr{H}}}^{d-1}(\{u_{\lambda }�=0\})�\ge c_{\mathrm{\Omega }}$, where $c_{\mathrm{\Omega }}$ is independent of $\lambda$. We also prove an almost sharp upper bound, namely, ${\mathcal{\mathscr{H}}}^{d-1}(\{u_{\lambda }�=0\})�\le C_{\mathrm{\Omega }}\lambda \log <!--FUNCTION\; APPLICATION-->(\lambda �+�e)$.

Let $E�\subset �B(1)�\subset {ℝ}^2$ be an ${\mathcal{\mathscr{H}}}^1$ measurable set with ${\mathcal{\mathscr{H}}}^1(E)�<�\infty$, and let $L�\subset {ℝ}^2$ be a line segment with ${\mathcal{\mathscr{H}}}^1(L)�={\mathcal{\mathscr{H}}}^1(E)$. It is not hard to see that $\mathrm{Fav}<!--FUNCTION\; APPLICATION-->(E)�\le \mathrm{Fav}<!--FUNCTION\; APPLICATION-->(L)$. We prove that in the case of near equality, that is,

the set $E$ can be covered by an $𝜖$-Lipschitz graph, up to a set of length $𝜖$. The dependence between $𝜖$ and $\delta$ is polynomial: in fact, the conclusions hold with $𝜖�=�C{\delta }^{1∕70}$ for an absolute constant $C�>0$.

Using the interpretation of the half-Laplacian on $S^1$ as the Dirichlet-to-Neumann operator for the Laplace equation on the ball $B$, we devise a classical approach to the heat flow for half-harmonic maps from $S^1$ to a closed target manifold $N�\subset {ℝ}^n$, recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the author’s 1985 results for the harmonic map heat flow of surfaces and in similar generality. When $N$ is a smoothly embedded, oriented closed curve $\Gamma �\subset {ℝ}^n$, the half-harmonic map heat flow may be viewed as an alternative gradient flow for a variant of the Plateau problem of disc-type minimal surfaces.

We obtain quantitative estimates on the fine structure of the singular set of the mutual boundary $\partial {\mathrm{\Omega }}^{\pm }$ for pairs of complementary domains ${\mathrm{\Omega }}^{+},{\mathrm{\Omega }}^{-}\subset {ℝ}^n$ which arise in a class of two-sided free boundary problems for harmonic measure. These estimates give new insight into the structure of the mutual boundary $\partial {\mathrm{\Omega }}^{\pm }$.

We study the differences of two consecutive eigenvalues ${\lambda }_i�-{\lambda }_{i-1}$, $i$ up to $2g�-2$, for the Laplacian on hyperbolic surfaces of genus $g$, and show that the supremum of such spectral gaps over the moduli space has infimum limit at least $\frac{1}{4}$ as the genus goes to infinity. A min-max principle for eigenvalues on degenerating hyperbolic surfaces is also established.

We continue our work on the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background Kähler metric of conical singularities along a divisor of simple normal crossings. As an application, we prove the short-time existence of the conical Kähler–Ricci flow with conical singularities along a divisor with simple normal crossings.

We study the problem of scattering by several strictly convex obstacles, with smooth boundary and satisfying a noneclipse condition. We show, in dimension 2 only, the existence of a spectral gap for the meromorphic continuation of the Laplace operator outside the obstacles. The proof of this result relies on a reduction to an open hyperbolic quantum map, achieved by Nonnenmacher et al. (Ann. ofMath. $(2)$179:1 (2014), 179–251). In fact, we obtain a spectral gap for this type of object, which also has applications in potential scattering. The second main ingredient of this article is a fractal uncertainty principle. We adapt the techniques of Dyatlov et al. (J. Amer. Math. Soc. 35:2 (2022), 361–465) to apply this fractal uncertainty principle in our context.

We obtain new characterizations of the Sobolev spaces ${Ẇ}^{1,p}({ℝ}^N)$ and the bounded variation space $\dot{\mathrm{BV}<!--FUNCTION\; APPLICATION-->}({ℝ}^N)$. The characterizations are in terms of the functionals ${\nu }_{\gamma }(E_{\lambda ,\gamma ∕p}[u])$, where

and the measure ${\nu }_{\gamma }$ is given by $d<!--FUNCTION\; APPLICATION-->{\nu }_{\gamma }(x,y)�=�|x�-�y{|}^{\gamma -N}d<!--FUNCTION\; APPLICATION-->xd<!--FUNCTION\; APPLICATION-->y$. We provide characterizations which involve the

We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $\mathbb{𝔻}$ in $ℂ$ into the unit ball ${\mathbb{𝔹}}^n$ of ${ℝ}^n$, $n�\ge 2$, at any point where the map is conformal. For $n�=2$ this generalizes the classical Schwarz–Pick lemma, and for $n�\ge 3$ it gives the optimal Schwarz–Pick lemma for conformal minimal discs $\mathbb{𝔻}�\to {\mathbb{𝔹}}^n$. This implies that conformal harmonic maps $M�\to {\mathbb{𝔹}}^n$ from any hyperbolic conformal surface are distance decreasing in the Poincaré metric on $M$ and the Cayley–Klein metric on the ball ${\mathbb{𝔹}}^n$, and the extremal maps are the conformal embeddings of the disc $\mathbb{𝔻}$ onto affine discs in ${\mathbb{𝔹}}^n$. Motivated by these results, we introduce an intrinsic pseudometric on any Riemannian manifold of dimension at least three by using conformal minimal discs, and we lay foundations of the corresponding hyperbolicity theory.

We prove that splash-like singularities cannot occur for sufficiently regular patch solutions to the generalized surface quasi-geostrophic equation on the plane or half-plane with parameter $\alpha �\le \frac{1}{4}$. This includes potential touches of more than two patch boundary segments in the same location, an eventuality that has not been excluded previously and presents nontrivial complications (in fact, if we do a priori exclude it, then our results extend to all $\alpha �\in �(0,1)$). As a corollary, we obtain an improved global regularity criterion for $H^3$ patch solutions when $\alpha �\le \frac{1}{4}$, namely that finite time singularities cannot occur while the $H^3$ norms of patch boundaries remain bounded.