Arachnology

Abstract We study spectral properties and geometric functional inequalities on Riemannian manifolds of dimension $$\ge 3$$ ≥ 3 with singularities. Of particular interest will be manifolds with (finite or countably many) conical singularities $$\{z_i\}_{i\in {\mathfrak {I}}}$$ { z i } i ∈ I in the neighborhood of which the largest lower bound for the Ricci curvature is $$\begin{aligned} k(x)\simeq K_i-\frac{s_i}{d^2(z_i,x)}. \end{aligned}$$ k ( x ) ≃ K i - s i d 2 ( z i , x ) . Thus none of the existing Bakry–Émery inequalities or curvature-dimension conditions apply. In particular, k does not belong to the Kato (or extended Kato) class, and (M, g) is not tamed in the sense of Erbar et al. (J Math Pures Appl 161: 1–69, 2022). Manifolds with such a singular Ricci bound (1) appear quite naturally. The prime examples are metric cones, for instance, $$M={{\mathbb {R}}}_+\times _r N$$ M = R + × r N with any $$(N,g^N)$$ ( N , g N ) satisfying $$\inf _{y\in N}\textrm{Ric}_y^N<(n-2)g^N$$ inf y ∈ N Ric y N < ( n - 2 ) g N , e.g. spheres $$N={{\mathbb {S}}}^{n-1}_R$$ N = S R n - 1 with radius $$R>1$$ R > 1 . For manifolds with such conical singularities we will prove a version of the Bakry–Émery inequality a novel Hardy inequality a spectral gap estimate. Related examples are provided by weighted spaces, e.g. $$M={{\mathbb {R}}}^n$$ M = R n with $$g=g^{Euclid}$$ g = g Euclid and $$m(dx)=|x|^\alpha d{\mathfrak {L}}^n(x)$$ m ( d x ) = | x | α d L n ( x ) for some $$\alpha \in {{\mathbb {R}}}$$ α ∈ R where the largest lower bound for Bakry–Émery Ricci tensor is given by $$ k(x)=-\frac{|\alpha |}{|x|^2}$$ k ( x ) = - | α | | x | 2 , and Grushin-type spaces $$M={{\mathbb {R}}}^j \times _f {{\mathbb {R}}}^{n-j}$$ M = R j × f R n - j with $$f(y)=|y|^{-\alpha }$$ f ( y ) = | y | - α for suitable $$\alpha >0$$ α > 0 , either with Riemannian volume measure or with Lebesgue measure, which admit lower Ricci bounds of the form $$k(y,z)=-\frac{C}{|y|^2}$$ k ( y , z ) = - C | y | 2 .

Abstract A new inequality for a nonlinear surface layer integral is proved for minimizers of causal variational principles. This inequality is applied to obtain a new proof of the positive mass theorem with volume constraint. Next, a positive mass theorem without volume constraint is stated and proved by introducing and using the concept of asymptotic alignment. Moreover, a positive quasilocal mass and a synthetic definition of scalar curvature are introduced in the setting of causal variational principles. Our notions and results are illustrated by the explicit examples of causal fermion systems constructed in ultrastatic spacetimes and the Schwarzschild spacetime. In these examples, the correspondence to the ADM mass and similarities to the Brown–York mass are worked out.

Abstract We consider the nonlinear Schrödinger equation with nonzero conditions at infinity in $$\textbf{R}^2$$ R 2 . We investigate the existence of traveling waves that are periodic in the direction transverse to the direction of propagation and minimize the energy when the momentum is kept fixed. We show that for any given value of the momentum, there is a critical value of the period such that traveling waves with period smaller than the critical value are one-dimensional, and those with larger periods depend on two variables.

Abstract Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\quad \hbox { in}\ \mathbb {R}^N_+$$ - L u = u p in R + N are obtained, $$-L$$ - L being any symmetric and stable linear operator, positively homogeneous of degree 2s, $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , whose spectral measure is absolutely continuous and positive only in a relative open set of the unit sphere of $$\mathbb {R}^N$$ R N . The results are sharp: $$u\equiv 0$$ u ≡ 0 is the only nonnegative supersolution in the subcritical regime $$1\le p\le \frac{N+s}{N-s}\,$$ 1 ≤ p ≤ N + s N - s , while nontrivial supersolutions exist, at least for some specific $$-L$$ - L , as soon as $$p>\frac{N+s}{N-s}$$ p > N + s N - s . The arguments used rely on a rescaled test function’s method, suitably adapted to such nonlocal setting with weak diffusion; they are quite general and also employed to obtain Liouville type results in the whole space.

Abstract We examine the possibility of finite-time blow-up of solutions to the fully parabolic quasilinear Keller–Segel model in a ball $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n with $$n\ge 2$$ n ≥ 2 . Previous results show that unbounded solutions exist for all $$m, q \in \mathbb {R}$$ m , q ∈ R with $$m-q<\frac{n-2}{n}$$ m - q < n - 2 n , which, however, are necessarily global in time if $$q \le 0$$ q ≤ 0 . It is expected that finite-time blow-up is possible whenever $$q > 0$$ q > 0 but in the fully parabolic setting this has so far only been shown when $$\max \{m, q\} \ge 1$$ max { m , q } ≥ 1 . In the present paper, we substantially extend these findings. Our main results for the two- and three-dimensional settings state that ( $$\star $$ ⋆ ) admits solutions blowing up in finite time if $$\begin{aligned} m-q<\frac{n-2}{n} \quad \text {and} \quad {\left\{ \begin{array}{ll} q< 2m & \text {if } n = 2, \\ q < 2m - \frac{2}{3} \text { or } m > \frac{2}{3} & \text {if } n = 3, \end{array}\right. } \end{aligned}$$ m - q < n - 2 n and q < 2 m if n = 2 , q < 2 m - 2 3 or m > 2 3 if n = 3 , that is, also for certain m, q with $$\max \{m, q\} < 1$$ max { m , q } < 1 . As a key new ingredient in our proof, we make use of (singular) pointwise upper estimates for u.

Abstract On a two-dimensional Riemannian manifold without boundary we consider the variational limit of a family of functionals given by the sum of two terms: a Ginzburg–Landau and a perimeter term. Our scaling allows low-energy states to be described by an order parameter which can have finitely many point singularities (vortex-like defects) of (possibly) fractional-degree connected by line discontinuities (string defects) of finite length. Our main result is a compactness and $$\Gamma $$ Γ -convergence theorem which shows how the coarse grained limit energy depends on the geometry of the manifold in driving the interaction between vortices and string defects.

Abstract We show that the volume of the boundary of a bounded Sobolev (p, q)-extension domain is zero when $$1\le q<p< \frac{qn}{(n-q)}.$$ 1 ≤ q < p < qn ( n - q ) .

Abstract We study a porous medium-type equation whose pressure is given by a nonlocal Lévy operator associated to a symmetric jump Lévy kernel. The class of nonlocal operators under consideration appears as a generalization of the classical fractional Laplace operator. For the class of Lévy operators, we construct weak solutions using a variational minimizing movement scheme. The lack of interpolation techniques is ensued by technical challenges that render our setting more challenging than the one known for fractional operators.

Abstract We study the Cahn-Hilliard equation with non-degenerate concentration-dependent mobility and logarithmic potential in two dimensions. We show that any weak solution is unique, exhibits propagation of uniform-in-time regularity, and stabilizes towards an equilibrium state of the Ginzburg-Landau free energy for large times. These results improve the state of the art dating back to a work by Barrett and Blowey. Our analysis relies on the combination of enhanced energy estimates, elliptic regularity theory and tools in critical Sobolev spaces.

Abstract Motivated by experiments and formal asymptotic expansions in the physics literature, Maor and Shachar (J. Elasticity 134 (2019), 149–173) studied the behaviour of a model elastic energy of maps between manifolds with incompatible metrics. For thin objects they analysed the scaling of the minimal elastic energy as a function of the thickness. In particular they showed that for maps from a ball of radius h in an oriented Riemannian manifold to Euclidean space, the infimum of a model elastic energy per unit volume scales like the fourth power of h and after rescaling one gets convergence to a quadratic expression in the curvature tensor R(p), where p denotes the centre of the ball. In this paper we show the same result for general compact oriented Riemannian targets with R(p) replaced by a suitable difference of the curvature tensors in the target and the domain, thus answering Open Question 1 in the paper by Maor and Shachar. The result extends to noncompact targets provided they satisfy a uniform regularity condition. A key idea in the proof is to use Lipschitz approximations to define a suitable notion of convergence.

Abstract We obtain new $$L_p$$ L p estimates for subsolutions to fully nonlinear equations. Based on our $$L_p$$ L p estimates, we further study several topics such as the third and fourth order derivative estimates for concave fully nonlinear equations, critical exponents of $$L_p$$ L p estimates and maximum principles, and the existence and uniqueness of solutions to fully nonlinear equations on the torus with free terms in the $$L_p$$ L p spaces or in the space of Radon measures.

In this paper, we investigate a system of equations derived from the $$\text {U}(1)\times \text {U}(1)$$ Abelian Chern-Simons model: $$\begin{aligned}\left\{ \begin{aligned} \Delta u&=\lambda \left( a(b-a)\textrm{e}^u-b(b-a)\textrm{e}^{\upsilon }+a^2\textrm{e}^{2u}-ab\textrm{e}^{2\upsilon }+b(b-a)\textrm{e}^{u+\upsilon } \right) +4\pi \sum \limits _{j=1}^{k_1}m_j\delta _{p_j},\\ \Delta \upsilon&=\lambda \left( -b(b-a)\textrm{e}^u+a(b-a)\textrm{e}^{\upsilon }-ab\textrm{e}^{2u}+a^2\textrm{e}^{2\upsilon }+b(b-a)\textrm{e}^{u+\upsilon } \right) +4\pi \sum \limits _{j=1}^{k_2}n_j\delta _{q_j}, \end{aligned} \right. \end{aligned}$$ on finite graphs. Here, $$\lambda >0$$ , $$b>a>0$$ , $$m_j>0\, (j=1,2,\ldots ,k_1)$$ , $$n_j>0\,(j=1,2,\ldots ,k_2)$$ , and $$\delta _{p}$$ denotes the Dirac delta mass at vertex p. We establish an iteration scheme and prove the existence of solutions. Additionally, we propose a novel method to derive the asymptotic behavior of solutions as $$\lambda $$ approaches infinity. This method is also applicable to the Chern-Simons system: $$\begin{aligned} \left\{ \begin{aligned} \Delta u&=\lambda \textrm{e}^{\upsilon }(\textrm{e}^{u}-1)+4\pi \sum \limits _{j=1}^{k_1}m_j\delta _{p_j},\\ \Delta \upsilon&=\lambda \textrm{e}^{u}(\textrm{e}^{\upsilon }-1)+4\pi \sum \limits _{j=1}^{k_2}n_j\delta _{q_j}, \end{aligned} \right. \end{aligned}$$ and the classical Chern-Simons equation: $$\begin{aligned} \Delta u=\lambda \textrm{e}^u(\textrm{e}^u-1)+4\pi \sum \limits _{j=1}^{N}\delta _{p_j}. \end{aligned}$$