Band 23, Heft 1

On a smooth bounded domain we study the Trudinger-Moser functional E α ( u ) ≔ ∫ Ω ( e α u 2 − 1 ) d x , u ∈ H 1 ( Ω ) {E}_{\alpha }\left(u):= \mathop{\int }\limits_{\Omega }({e}^{\alpha {u}^{2}}-1){\rm{d}}x,\hspace{1.0em}u\in {H}^{1}\left(\Omega ) for α ∈ ( 0 , 2 π ) \alpha \in \left(0,2\pi ) and its restriction E α ∣ Σ λ {E}_{\alpha }{| }_{{\Sigma }_{\lambda }} , where Σ λ ≔ u ∈ H 1 ( Ω ) ∣ ∫ Ω ( ∣ ∇ u ∣ 2 + λ u 2 ) d x = 1 {\Sigma }_{\lambda }:= \left\{u\in {H}^{1}\left(\Omega )| {\int }_{\Omega }(| \nabla u{| }^{2}+\lambda {u}^{2}){\rm{d}}x=1\right\} for λ > 0 \lambda \gt 0 . By applying the asymptotic analysis and the variational method, we obtain asymptotic behavior of critical points of E α ∣ Σ λ {E}_{\alpha }{| }_{{\Sigma }_{\lambda }} both as λ → 0 \lambda \to 0 and as λ → + ∞ \lambda \to +\infty . In particular, we prove that when α \alpha is sufficiently small, maximizers for sup u ∈ Σ λ E α ( u ) {\sup }_{u\in {\Sigma }_{\lambda }}{E}_{\alpha }\left(u) tend to 0 in C ( Ω ¯ ) C\left(\overline{\Omega }) as λ → + ∞ \lambda \to +\infty .

We consider the existence and nonexistence of the positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: − Δ u = ∣ u ∣ 2 ∗ − 2 u + λ u + μ u log u 2 x ∈ Ω , u = 0 x ∈ ∂ Ω , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}-\Delta u={| u| }^{{2}^{\ast }-2}u+\lambda u+\mu u\log {u}^{2}\hspace{1.0em}& x\in \Omega ,\\ u=0\hspace{1.0em}& x\in \partial \Omega ,\end{array}\right. where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded open domain, λ , μ ∈ R \lambda ,\mu \in {\mathbb{R}} , N ≥ 3 N\ge 3 and 2 ∗ ≔ 2 N N − 2 {2}^{\ast }:= \frac{2N}{N-2} is the critical Sobolev exponent for the embedding H 0 1 ( Ω ) ↪ L 2 ∗ ( Ω ) {H}_{0}^{1}\left(\Omega )\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{{2}^{\ast }}\left(\Omega ) . The uncertainty of the sign of s log s 2 s\log {s}^{2} in ( 0 , + ∞ ) \left(0,+\infty ) has some interest in itself. We will show the existence of positive ground state solution, which is of mountain pass type provided λ ∈ R , μ > 0 \lambda \in {\mathbb{R}},\mu \gt 0 and N ≥ 4 N\ge 4 . While the case of μ < 0 \mu \lt 0 is thornier. However, for N = 3 , 4 N=3,4 , λ ∈ ( − ∞ , λ 1 ( Ω ) ) \lambda \in \left(-\infty ,{\lambda }_{1}\left(\Omega )) , we can also establish the existence of positive solution under some further suitable assumptions. A nonexistence result is also obtained for μ < 0 \mu \lt 0 and − ( N − 2 ) μ 2 + ( N − 2 ) μ 2 log − ( N − 2 ) μ 2 + λ − λ 1 ( Ω ) ≥ 0 -\frac{\left(N-2)\mu }{2}+\frac{\left(N-2)\mu }{2}\log \left(-\frac{\left(N-2)\mu }{2}\right)+\lambda -{\lambda }_{1}\left(\Omega )\ge 0 if N ≥ 3 N\ge 3 . Comparing with the results in the study by Brézis and Nirenberg ( Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents , Comm. Pure Appl. Math. 36 (1983), 437–477), some new interesting phenomenon occurs when the parameter μ \mu on logarithmic perturbation is not zero.

The natural consequence of the existence of different kinds of chaos is the study of their mutual dependence and the relationship between these concepts and the entropy of systems. This observation also applies to the local approach to this issue. In this article, we will focus on this problem in the context of “points focusing chaos.” We aim to show their mutual independence by considering the sets of appropriate periodic dynamical systems in the space of discrete dynamical systems.

In this article, we study the sharp profiles of positive solutions to the diffusive logistic equation. By employing parameters and analyzing the corresponding perturbation equations, we find the effects of boundary and spatial heterogeneity on the positive solutions. The main results exhibit the sharp effects between boundary conditions and linear/nonlinear spatial heterogeneities on positive solutions.

In this article, we prove that, generically in the sense of domain variations, any solution to a nonlinear eigenvalue problem is either nondegenerate or the Crandall-Rabinowitz transversality condition that is satisfied. We then deduce that, generically, the unbounded Rabinowitz continuum of solutions is a simple analytic curve. The global bifurcation diagram resembles the classic model case of the Gel’fand problem in two dimensions.

This article is concerned with the stationary problem for a prey-predator model with prey-taxis/predator-taxis under homogeneous Dirichlet boundary conditions, where the interaction is governed by a Beddington-DeAngelis functional response. We make a detailed description of the global bifurcation structure of coexistence states and find the ranges of parameters for which there exist coexistence states. At the same time, some sufficient conditions for the nonexistence of coexistence states are also established. Our method of analysis uses the idea developed by Cintra et al. ( Unilateral global bifurcation for a class of quasilinear elliptic systems and applications , J. Differential Equations 267 (2019), 619–657). Our results indicate that the presence of prey-taxis/predator-taxis makes mathematical analysis more difficult, and the Beddington-DeAngelis functional response leads to some different phenomena.

In this article, we are concerned with multiple solutions of Schrödinger-Choquard-Kirchhoff equations involving the fractional p p -Laplacian and Hardy-Littlewood-Sobolev critical exponents in R N {{\mathbb{R}}}^{N} . We classify the multiplicity of the solutions in accordance with the Kirchhoff term M ( ⋅ ) M\left(\cdot ) and different ranges of q q shown in the nonlinearity f ( x , ⋅ ) f\left(x,\cdot ) by means of the variational methods and Krasnoselskii’s genus theory. As an immediate consequence, some recent related results have been improved and extended.

Let I I be a bounded interval of R {\mathbb{R}} and λ 1 ( I ) {\lambda }_{1}\left(I) denote the first eigenvalue of the nonlocal operator ( − Δ ) 1 4 {(-\Delta )}^{\tfrac{1}{4}} with the Dirichlet boundary. We prove that for any 0 ⩽ α < λ 1 ( I ) 0\leqslant \alpha \lt {\lambda }_{1}(I) , there holds sup u ∈ W 0 1 2 , 2 ( I ) , ‖ ( − Δ ) 1 4 u ‖ 2 2 − α ∥ u ∥ 2 2 ≤ 1 ∫ I e π u 2 d x < + ∞ , \mathop{\sup }\limits_{u\in {W}_{0}^{\frac{1}{2},2}(I),\Vert {\left(-\Delta )}^{\tfrac{1}{4}}u{\Vert }_{2}^{2}-\alpha {\parallel u\parallel }_{2}^{2}\le 1}\mathop{\int }\limits_{I}{e}^{\pi {u}^{2}}{\rm{d}}x\lt +\infty , and the supremum can be attained. The method is based on concentration-compactness principle for fractional Trudinger-Moser inequality, blow-up analysis for fractional elliptic equation with the critical exponential growth and harmonic extensions.

The primary objective of this article is to analyze the existence of infinitely many radial p p - k k -convex solutions to the boundary blow-up p p - k k -Hessian problem σ k ( λ ( D i ( ∣ D u ∣ p − 2 D j u ) ) ) = H ( ∣ x ∣ ) f ( u ) in Ω , u = + ∞ on ∂ Ω . {\sigma }_{k}\left(\lambda \left({D}_{i}\left({| Du| }^{p-2}{D}_{j}u)))=H\left(| x| )f\left(u)\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{0.33em}u=+\infty \hspace{0.33em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega . Here, k ∈ { 1 , 2 , … , N } k\in \left\{1,2,\ldots ,N\right\} , σ k ( λ ) {\sigma }_{k}\left(\lambda ) is the k k -Hessian operator, and Ω \Omega is a ball in R N ( N ≥ 2 ) {{\mathbb{R}}}^{N}\hspace{0.33em}\left(N\ge 2) . Our methods are mainly based on the sub- and super-solutions method.

In the present article, we prove the existence and uniqueness of smooth solutions to an anisotropic L p {L}_{p} Minkowski problem for the log-concave measure. Our proof of the existence is based on the well-known continuous method whose crucial factor is the a priori bounds of an auxiliary problem. The uniqueness is based on a maximum principle argument. It is worth mentioning that apart from the C 2 {C}^{2} bounds of solutions, the C 1 {C}^{1} bounds of solutions also need some efforts since the convexity of S S cannot be used directly, which is one of great difference between the classical and the anisotropic versions. Moreover, our result can be seen as an attempt to get new results on the geometric analysis of log-concave measure.

In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term , J. Math. Pures Appl. 128 (2019), pp. 183–212], we proved the existence of a threshold κ ∗ > 0 {\kappa }^{\ast }\gt 0 such that the elliptic problem for an inhomogeneous elliptic equation − Δ u + u = u p + κ μ -\Delta u+u={u}^{p}+\kappa \mu in R N {{\bf{R}}}^{N} possesses a positive minimal solution decaying at the space infinity if and only if 0 < κ ≤ κ ∗ 0\lt \kappa \le {\kappa }^{\ast } . Here, N ≥ 2 N\ge 2 , μ \mu is a nontrivial nonnegative Radon measure in R N {{\bf{R}}}^{N} with a compact support, and p > 1 p\gt 1 is in the Joseph-Lundgren subcritical case. In this article, we prove the existence of nonminimal positive solutions to the elliptic problem. Our arguments are also applicable to inhomogeneous semilinear elliptic equations with exponential nonlinearity.

In this article, we consider the following Schrödinger-Poisson problem: − ε 2 Δ u + V ( y ) u + Φ ( y ) u = ∣ u ∣ p − 1 u , y ∈ R 3 , − Δ Φ ( y ) = u 2 , y ∈ R 3 , \left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V(y)u+\Phi (y)u={| u| }^{p-1}u,& y\in {{\mathbb{R}}}^{3},\\ -\Delta \Phi (y)={u}^{2},& y\in {{\mathbb{R}}}^{3},\end{array}\right. where ε > 0 \varepsilon \gt 0 is a small parameter, 1 < p < 5 1\lt p\lt 5 , and V ( y ) V(y) is a potential function. We construct multi-peak solution concentrating at the critical points of V ( y ) V(y) through the Lyapunov-Schmidt reduction method. Moreover, by using blow-up analysis and local Pohozaev identities, we prove that the multi-peak solution we construct is non-degenerate. To our knowledge, it seems be the first non-degeneracy result on the Schödinger-Poisson system.

Our main purpose is to establish Gagliardo-Nirenberg-type inequalities using fractional homogeneous Sobolev spaces and homogeneous Besov spaces. In particular, we extend some of the results obtained by the authors in previous studies.

We are concerned with the following coupled nonlinear Schrödinger system: − Δ u + u + ∫ ∣ x ∣ ∞ h ( s ) s u 2 ( s ) d s + h 2 ( ∣ x ∣ ) ∣ x ∣ 2 u = ∣ u ∣ 2 p − 2 u + b ∣ v ∣ p ∣ u ∣ p − 2 u , x ∈ R 2 , − Δ v + ω v + ∫ ∣ x ∣ ∞ g ( s ) s v 2 ( s ) d s + g 2 ( ∣ x ∣ ) ∣ x ∣ 2 v = ∣ v ∣ 2 p − 2 v + b ∣ u ∣ p ∣ v ∣ p − 2 v , x ∈ R 2 , \left\{\begin{array}{l}-\Delta u+u+\left(\underset{| x| }{\overset{\infty }{\displaystyle \int }}\frac{h\left(s)}{s}{u}^{2}\left(s){\rm{d}}s+\frac{{h}^{2}\left(| x| )}{{| x| }^{2}}\right)u={| u| }^{2p-2}u+b{| v| }^{p}{| u| }^{p-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{2},\hspace{1.0em}\\ -\Delta v+\omega v+\left(\underset{| x| }{\overset{\infty }{\displaystyle \int }}\frac{g\left(s)}{s}{v}^{2}\left(s){\rm{d}}s+\frac{{g}^{2}\left(| x| )}{{| x| }^{2}}\right)v={| v| }^{2p-2}v+b{| u| }^{p}{| v| }^{p-2}v,\hspace{1em}x\in {{\mathbb{R}}}^{2},\hspace{1.0em}\end{array}\right. where ω , b > 0 \omega ,b\gt 0 , p > 1 p\gt 1 . By virtue of the variational approach, we show the existence of nontrivial ground-state solutions depending on the parameters involved. Precisely, the aforementioned system admits a positive ground-state solution if p > 3 p\gt 3 and b > 0 b\gt 0 large enough or if p ∈ ( 2 , 3 ] p\in \left(2,3] and b > 0 b\gt 0 small.

In this work, we analyze the asymptotic behavior of a class of quasilinear elliptic equations defined in oscillating ( N + 1 ) \left(N+1) -dimensional thin domains (i.e., a family of bounded open sets from R N + 1 {{\mathbb{R}}}^{N+1} , with corrugated bounder, which degenerates to an open bounded set in R N {{\mathbb{R}}}^{N} ). We also allow monotone nonlinear boundary conditions on the rough border whose magnitude depends on the squeezing of the domain. According to the intensity of the roughness and a reaction coefficient term on the nonlinear boundary condition, we obtain different regimes establishing effective homogenized limits in N N -dimensional open bounded sets. In order to do that, we combine monotone operator analysis techniques and the unfolding method used to deal with asymptotic analysis and homogenization problems.

This article deals with existence of solutions to the following fractional p p -Laplacian system of equations: ( − Δ p ) s u = ∣ u ∣ p s * − 2 u + γ α p s * ∣ u ∣ α − 2 u ∣ v ∣ β in Ω , ( − Δ p ) s v = ∣ v ∣ p s * − 2 v + γ β p s * ∣ v ∣ β − 2 v ∣ u ∣ α in Ω , \left\{\begin{array}{l}{\left(-{\Delta }_{p})}^{s}u={| u| }^{{p}_{s}^{* }-2}u+\frac{\gamma \alpha }{{p}_{s}^{* }}{| u| }^{\alpha -2}u{| v| }^{\beta }\hspace{0.33em}\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{1.0em}\\ {\left(-{\Delta }_{p})}^{s}v={| v| }^{{p}_{s}^{* }-2}v+\frac{\gamma \beta }{{p}_{s}^{* }}{| v| }^{\beta -2}v{| u| }^{\alpha }\hspace{0.33em}\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{1.0em}\end{array}\right. where s ∈ ( 0 , 1 ) s\in \left(0,1) , p ∈ ( 1 , ∞ ) p\in \left(1,\infty ) with N > s p N\gt sp , α , β > 1 \alpha ,\beta \gt 1 such that α + β = p s * ≔ N p N − s p \alpha +\beta ={p}_{s}^{* }:= \frac{Np}{N-sp} and Ω = R N \Omega ={{\mathbb{R}}}^{N} or smooth bounded domains in R N {{\mathbb{R}}}^{N} . When Ω = R N \Omega ={{\mathbb{R}}}^{N} and γ = 1 \gamma =1 , we show that any ground state solution of the aforementioned system has the form ( λ U , τ λ V ) \left(\lambda U,\tau \lambda V) for certain τ > 0 \tau \gt 0 and U U and V V are two positive ground state solutions of ( − Δ p ) s u = ∣ u ∣ p s * − 2 u {\left(-{\Delta }_{p})}^{s}u={| u| }^{{p}_{s}^{* }-2}u in R N {{\mathbb{R}}}^{N} . For all γ > 0 \gamma \gt 0 , we establish existence of a positive radial solution to the aforementioned system in balls. When Ω = R N \Omega ={{\mathbb{R}}}^{N} , we also establish existence of positive radial solutions to the aforementioned system in various ranges of γ \gamma .

In this article, we study the uniqueness of positive symmetric solutions of the following mean curvature problem in Euclidean space: (P) u ′ 1 + ∣ u ′ ∣ 2 ′ + h ( x ) f ( u ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , \left\{\begin{array}{l}{\left(\frac{u^{\prime} }{\sqrt{1+{| u^{\prime} | }^{2}}}\right)}^{^{\prime} }+h\left(x)f\left(u)=0,\hspace{1em}-1\lt x\lt 1,\hspace{1.0em}\\ u\left(-1)=u\left(1)=0,\hspace{1.0em}\end{array}\right. where h ∈ C 1 ( [ − 1 , 1 ] ) h\in {C}^{1}\left(\left[-1,1]) and f ∈ C 1 ( [ 0 , ∞ ) ; [ 0 , ∞ ) ) f\in {C}^{1}\left(\left[0,\infty );\hspace{0.33em}\left[0,\infty )) . Under suitable conditions on h h and monotone condition on f ( s ) s \frac{f\left(s)}{s} , by introducing a modified Picone-type identity, we prove that the problem has at most one positive symmetric solution.

By combining monotonicity theory related to the parametric version of the Browder-Minty theorem with fixed point arguments, hybrid existence results for a system of two operator equations are obtained. Applications are given to a system of boundary value problems with mixed nonlocal and Dirichlet conditions.

In this article, we study the Musielak-Orlicz-Gauss image problem based on the Gauss curvature flow in Li et al. We deal with some cases in which there is no uniform estimate for the Gauss curvature flow. By the use of the topological method in Guang et al., a special initial condition is chosen such that the Gauss curvature flow converges to a solution of the Musielak-Orlicz-Gauss image problem.

We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ \Gamma in a Cartan-Hadamard manifold M M . In particular, we show that the first mean curvature integral of a convex hypersurface γ \gamma nested inside Γ \Gamma cannot exceed that of Γ \Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ \Gamma in terms of the volume it bounds in M M in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ \gamma is parallel to Γ \Gamma , or M M has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.

In this article, we prove the C 1,1 estimate for solutions of prescribed curvature measure problems when the prescribed function may touch zero somewhere.

This article concerns the L p {L}_{p} Minkowski problem for q -capacity. We consider the case p ≥ 1 p\ge 1 and 1 < q < n 1\lt q\lt n in the smooth category by a kind of curvature flow, which converges smoothly to the solution of a Monge-Ampére type equation. We show the existence of smooth solution to the problem for p ≥ n p\ge n . We also provide a proof for the weak solution to the problem when p ≥ 1 p\ge 1 , which has been obtained by Zou and Xiong.

We survey some ideas regarding the application of the Aleksandrov reflection method in partial differential equation to extrinsic geometric flows of Euclidean hypersurfaces. In this survey, we mention some related and important recent developments of others on the convergence of noncontracting flows and construction and classification of ancient flows.

In this article, we revisit previous Pogorelov-type interior and global second derivative estimates of N. S. Trudinger, F. Jiang, and J. Liu for solutions of Monge-Ampère-type partial differential equations. Taking account of recent strict convexity regularity results of Guillen-Kitagawa and Rankin and following our earlier work in the optimal transportation case, we remove the monotonicity assumptions in the more general case of generated Jacobian equations and consequently in the subsequent application to classical solvability and global regularity for second boundary value problems.

Chord measures are newly discovered translation-invariant geometric measures of convex bodies in R n {{\mathbb{R}}}^{n} , in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing the L p {L}_{p} chord measures is called the L p {L}_{p} chord Minkowski problem in the L p {L}_{p} Brunn-Minkowski theory, which includes the L p {L}_{p} Minkowski problem as a special case. This article solves the L p {L}_{p} chord Minkowski problem when p > 1 p\gt 1 and the symmetric case of 0 < p < 1 0\lt p\lt 1 .

We observe that the k k -dimensional width of an n n -ball in a space form is given by the area of an equatorial k k -ball. We also discuss the relationship between widths and lower bounds for the area of a free boundary minimal submanifold in a space form ball.

In this article, we prove the existence of smooth approximations of twisted Kähler-Einstein metrics using the variational method.

In this article, we consider the homogeneous complex k k -Hessian equation in an exterior domain C n ⧹ Ω {{\mathbb{C}}}^{n}\setminus \Omega . We prove the existence and uniqueness of the C 1 , 1 {C}^{1,1} solution by constructing approximating solutions. The key point for us is to establish the uniform gradient estimate and the second-order estimate.

In this article, we prove a Carleman inequality on a product manifold M × R M\times {\mathbb{R}} . As applications, we prove that (1) a periodic harmonic function on R 2 {{\mathbb{R}}}^{2} that decays faster than all exponential rate in one direction must be constant 0, (2) a periodic minimal hypersurface in R 3 {{\mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rate in one direction must be a hyperplane, and (3) a periodic translator in R 3 {{\mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rates in one direction must be a translating hyperplane.

We consider the asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity and fulfill previous results under faster convergence rate by Bao et al. [ Monge-Ampère equation on exterior domains , Calc. Var PDE. 52 (2015), 39–63]. Different from known results, we obtain the limit of Hessian and/or gradient of solution at infinity relying on the convergence rate. The basic idea is to use a revised level set method, the spherical harmonic expansion, and the iteration method.

We make rigorous and old idea of using mean curvature flow to prove a theorem of Richard Hamilton on the compactness of proper hypersurfaces with pinched, bounded curvature.

This article introduces a functional generalizing Perelman’s weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well defined on a wide class of noncompact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and the Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.

Given r 0 > 0 {r}_{0}\gt 0 , I ∈ N ∪ { 0 } I\in {\mathbb{N}}\cup \left\{0\right\} , and K 0 , H 0 ≥ 0 {K}_{0},{H}_{0}\ge 0 , let X X be a complete Riemannian 3-manifold with injectivity radius Inj ( X ) ≥ r 0 \hspace{0.1em}\text{Inj}\hspace{0.1em}\left(X)\ge {r}_{0} and with the supremum of absolute sectional curvature at most K 0 {K}_{0} , and let M ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}X be a complete immersed surface of constant mean curvature H ∈ [ 0 , H 0 ] H\in \left[0,{H}_{0}] and with index at most I I . We will obtain geometric estimates for such an M ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}X as a consequence of the hierarchy structure theorem. The hierarchy structure theorem (Theorem 2.2) will be applied to understand global properties of M ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}X , especially results related to the area and diameter of M M . By item E of Theorem 2.2, the area of such a noncompact M ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}X is infinite. We will improve this area result by proving the following when M M is connected; here, g ( M ) g\left(M) denotes the genus of the orientable cover of M M : (1) There exists C 1 = C 1 ( I , r 0 , K 0 , H 0 ) > 0 {C}_{1}={C}_{1}\left(I,{r}_{0},{K}_{0},{H}_{0})\gt 0 , such that Area ( M ) ≥ C 1 ( g ( M ) + 1 ) {\rm{Area}}\left(M)\ge {C}_{1}\left(g\left(M)+1) . (2) There exist C > 0 C\gt 0 , G ( I ) ∈ N G\left(I)\in {\mathbb{N}} independent of r 0 , K 0 , H 0 {r}_{0},{K}_{0},{H}_{0} , and also C C independent of I I such that if g ( M ) ≥ G ( I ) g\left(M)\ge G\left(I) , then Area ( M ) ≥ C ( max 1 , 1 r 0 , K 0 , H 0 ) 2 ( g ( M ) + 1 ) {\rm{Area}}\left(M)\ge \frac{C}{{\left(\max \left\{1,\frac{1}{{r}_{0}},\sqrt{{K}_{0}},{H}_{0}\right\}\right)}^{2}}\left(g\left(M)+1) . (3) If the scalar curvature ρ \rho of X X satisfies 3 H 2 + 1 2 ρ ≥ c 3{H}^{2}+\frac{1}{2}\rho \ge c in X X for some c > 0 c\gt 0 , then there exist A , D > 0 A,D\gt 0 depending on c , I , r 0 , K 0 , H 0 c,I,{r}_{0},{K}_{0},{H}_{0} such that Area ( M ) ≤ A {\rm{Area}}\left(M)\le A and Diameter ( M ) ≤ D {\rm{Diameter}}\left(M)\le D . Hence, M M is compact and, by item 1, g ( M ) ≤ A / C 1 − 1 g\left(M)\le A\hspace{0.1em}\text{/}\hspace{0.1em}{C}_{1}-1 .

In this article, we introduce a notion of capillary Schwarz symmetrization in the half-space. It can be viewed as the counterpart of the classical Schwarz symmetrization in the framework of capillary problem in the half-space. A key ingredient is a special anisotropic gauge, which enables us to transform the capillary symmetrization to the convex symmetrization introduced in Alvino et al. https:/doi.org/10.1016/S0294-1449(97)80147-3.

In this note, we establish the global C 3 , α {C}^{3,\alpha } regularity for potential functions in optimal transportation between hypercubes in R n {{\mathbb{R}}}^{n} for n ≥ 3 n\ge 3 . When n = 2 n=2 , the result was proved by Jhaveri. The C 3 , α {C}^{3,\alpha } regularity is also optimal due to a counterexample in the study by Jhaveri.

We consider a nonlinear Neumann problem, with periodic oscillation in the elliptic operator and on the boundary condition. Our focus is on problems posed in half-spaces, but with general normal directions that may not be parallel to the directions of periodicity. As the frequency of the oscillation grows, quantitative homogenization results are derived. When the homogenized operator is rotation-invariant, we prove the Hölder continuity of the homogenized boundary data. While we follow the outline of Choi and Kim ( Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data , Journal de Mathématiques Pures et Appliquées 102 (2014), no. 2, 419–448), new challenges arise due to the presence of tangential derivatives on the boundary condition in our problem. In addition, we improve and optimize the rate of convergence within our approach. Our results appear to be new even for the linear oblique problem.

The goal of this article is to give a new proof of the wave trace formula proved by Richard Melrose in an impressive article. This trace formula is an extension of the Chazarain-Duistermaat-Guillemin trace formula (denoted as “CDG trace formula” in this article) to the case of a sub-Riemannian Laplacian on a 3D contact closed manifold. The proof uses a normal form constructed in previous papers, following the pioneering work of Melrose to reduce the case of the invariant Laplacian on the 3D-Heisenberg group. We need also the propagation of singularities results of the works of Ivrii, Lascar, and Melrose.

We investigate the rigidity of global minimizers u ≥ 0 u\ge 0 of the Alt-Phillips functional involving negative power potentials ∫ Ω ( ∣ ∇ u ∣ 2 + u − γ χ { u > 0 } ) d x , γ ∈ ( 0 , 2 ) , \mathop{\int }\limits_{\Omega }(| \nabla u{| }^{2}+{u}^{-\gamma }{\chi }_{\left\{u\gt 0\right\}}){\rm{d}}x,\hspace{1.0em}\gamma \in \left(0,2), when the exponent γ \gamma is close to the extremes of the admissible values. In particular, we show that global minimizers in R n {{\mathbb{R}}}^{n} are one-dimensional if γ \gamma is close to 2 and n ≤ 7 n\le 7 , or if γ \gamma is close to 0 and n ≤ 4 n\le 4 .

In this note, we extend the regularity theory for monotone measure-preserving maps, also known as optimal transports for the quadratic cost optimal transport problem, to the case when the support of the target measure is an arbitrary convex domain and, on the low-regularity end, between domains carrying certain invariant measures.

Let u u be a nontrivial harmonic function in a domain D ⊂ R d D\subset {{\mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} -Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } \left\{X\in \overline{D}:u\left(X)=0=| \nabla u\left(X)| \right\} , has finite ( d − 2 ) \left(d-2) -dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} -Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{\mathcal{ {\mathcal H} }}}^{d-2} -measures.

We are interested in sharp functional inequalities for the coherent state transform related to the Wehrl conjecture and its generalizations. This conjecture was settled by Lieb in the case of the Heisenberg group, Lieb and Solovej for SU(2), and Kulikov for SU(1, 1) and the affine group. In this article, we give alternative proofs and characterize, for the first time, the optimizers in the general case. We also extend the recent Faber-Krahn-type inequality for Heisenberg coherent states, due to Nicola and Tilli, to the SU(2) and SU(1, 1) cases. Finally, we prove a family of reverse Hölder inequalities for polynomials, conjectured by Bodmann.

We establish the Caccioppoli inequality, a reverse Hölder inequality in the spirit of the classic estimate of Meyers, and construct the fundamental solution for linear elliptic differential equations of order 2 m 2m with certain lower order terms.

We will show that if a gradient shrinking Ricci soliton has an approximate symmetry on one scale, this symmetry propagates to larger scales. This is an example of the shrinker principle which roughly states that information radiates outwards for shrinking solitons.

Let 1 < p < ∞ 1\lt p\lt \infty and suppose that we are given a function f f defined on the leaves of a weighted tree. We would like to extend f f to a function F F defined on the entire tree, so as to minimize the weighted W 1 , p {W}^{1,p} -Sobolev norm of the extension. An easy situation is when p = 2 p=2 , where the harmonic extension operator provides such a function F F . In this note, we record our analysis of the particular case of a radially symmetric binary tree, which is a complete, finite, binary tree with weights that depend only on the distance from the root. Neither the averaging operator nor the harmonic extension operator work here in general. Nevertheless, we prove the existence of a linear extension operator whose norm is bounded by a constant depending solely on p p . This operator is a variant of the standard harmonic extension operator, and in fact, it is harmonic extension with respect to a certain Markov kernel determined by p p and by the weights.

In this study, the solution of the Neumann problem associated with the CR Yamabe operator on a subset Ω \Omega of the CR manifold S 3 {{\mathbb{S}}}^{3} bounded by the Clifford torus Σ \Sigma is discussed. The Yamabe-type problem of finding a contact form on Ω \Omega which has zero Tanaka-Webster scalar curvature and for which Σ \Sigma has a constant p p -mean curvature is also discussed.

We show that there is a one-to-one correspondence between solutions to the Poisson-landscape equations and the reduced Hartree-Fock equations in the semi-classical limit at low temperature. Moreover, we prove that the difference between the two corresponding solutions is small by providing explicit estimates.

We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the noncommutative nilpotent setting. In particular, we present what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.

In this article, we first apply the method of combining the interpolation theorem and weak-type estimate developed in Chen et al. to derive the Hardy-Littlewood-Sobolev inequality with an extended Poisson kernel. By using this inequality and weighted Hardy inequality, we further obtain the Stein-Weiss inequality with an extended Poisson kernel. For the extremal problem of the corresponding Stein-Weiss inequality, the presence of double-weighted exponents not being necessarily nonnegative makes it impossible to obtain the desired existence result through the usual technique of symmetrization and rearrangement. We then adopt the concentration compactness principle of double-weighted integral operator, which was first used by the authors in Chen et al. to overcome this difficulty and obtain the existence of the extremals. Finally, the regularity of the positive solution for integral system related with the extended kernel is also considered in this article. Our regularity result also avoids the nonnegativity condition of double-weighted exponents, which is a common assumption in dealing with the regularity of positive solutions of the double-weighted integral systems in the literatures.

By applying the gluing method, we construct infinitely many axial symmetric singular positive solutions to the Lane-Emden equation: Δ H u + u p = 0 , in H n \ { 0 } {\Delta }_{{\mathbb{H}}}u+{u}^{p}=0\left,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{H}}}^{n}\backslash \left\{0\right\} on the Heisenberg group H n {{\mathbb{H}}}^{n} , where n > 1 , Q ⁄ ( Q − 4 ) < p < p J L ( Q − 2 ) n\gt 1,Q/\left(Q-4)\lt p\lt {p}_{JL}\left(Q-2) , and Q = 2 n + 2 Q=2n+2 is the homogeneous dimension of H n {{\mathbb{H}}}^{n} .