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In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number $\chi(M)

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if and only if there are only finitely many spherical singularities. More generally, using the solution of the mean convex neighborhood conjecture for neck singularities, we establish a local version of this equivalence for neck singularities in arbitrary dimension. In particular, we see that the ancient ovals occur as limit flows if and only if there is a sequence of spherical singularities converging to a neck singularity., Comment: 5 pages

We establish a spatial gradient maximum principle for classical solutions to the initial and Neumann boundary value problem of some quasilinear parabolic equations on smooth convex domains., To appear in Proc. Amer. Math. Soc

Let $M$ be a Kobayashi hyperbolic homogenous manifold. Let $\mathcal F$ be a holomorphic foliation on $M$ invariant under a transitive group $G$ of biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a holomorphic $G$-equivariant submersion $\pi \colon M \to N$ onto a $G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in $\mathbb C^n$, then the fixed point set of $\mathcal Q$ is either empty or a connected complex submanifold of $D$., Comment: final version

By means of dual convex bodies, we obtain regularity of solutions to the expanding Gauss curvature flows with homogeneity degrees $-p$, $0, Comment: To appear in Proc. Amer. Math. Soc

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS

We prove the following subadditive property of the error function: \[ erf ( x ) = 2 π ∫ 0 x e − t 2 d t ( x ∈ R ) . \mbox {erf}\,(x)=\frac {2}{\sqrt {\pi }}\int _0^x e^{-t^2}dt \quad {(x\in \mathbf {R})}. \] Let a a and b b be real numbers. The inequality \[ erf ( ( x + y ) a ) b > erf ( x a ) b + erf ( y a ) b \mbox {erf}\,\bigl ((x+y)^a\bigr )^b> \mbox {erf}\,(x^a)^b + \mbox {erf}\,(y^a)^b \] holds for all positive real numbers x x and y y if and only if a b ≤ 1 ab\leq 1 .

We show that Caratheodory’s conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f : R → R whose gradient decays uniformly faster than 1/r. The divergence theorem then yields a pair of integral equations for the normal curvatures of these graphs, which establish some weaker forms of the conjecture. In particular, we show that there are uncountably many principal lines in the graph of f whose projection into R are parallel to any given direction.

For continuous differential inclusions the classical bang-bang property is known to fail, yet a weak form of it is established here, in the case where the right hand side is a multifunction whose values are closed convex and bounded sets with nonempty interior contained in a reflexive and separable Banach space. Our approach is based on the Baire category method.

A construction of separated sequences in the unit sphere of a Banach space is given. If a space X admits an equivalent nearly uniformly convex norm or co is not finitely representable in X, then lower bounds for separation constants of sequences are strictly greater than 1. This gives a partial answer to a problem posed by J. Diestel.

Consider a unit-resistive plate in the shape of a regular polygon with 2 n 2n sides, in which even-numbered sides are wired to electrodes and odd-numbered sides are insulated. The response matrix, or Dirichlet-to-Neumann map, allows one to compute the currents flowing through the electrodes when they are held at specified voltages. We show that the entries of the response matrix of the regular 2 n 2n -gon are given by the differences of cotangents of evenly spaced angles, and we describe some connections with the limiting distributions of certain random spanning forests.

If X is a finite dimensional real normed space, C is a closed convex subset of X and f: C → C is nonexpansive with respect to the norm on X, then we show that either f has a fixed point in C or there is a linear functional ϕ ∈ X * such that lim k→∞ ϕ(f k (x)) = ∞ for all x ∈ C.

We prove that generalized convex radial solutions to the generalized Monge-Ampère equation det D 2 u = f ( | x | 2 / 2 , u , | ∇ u | 2 / 2 ) \det D^2u = f(|x|^2/2,u,|\nabla u|^2/2) with f f smooth are always smooth away from the origin. Moreover, we characterize the global smoothness of these solutions in terms of the order of vanishing of f f at the origin.

Let 0 < a n ,b n ,c n such that a n = b n c n . If a = lim a→∞ a n , and {b n } and {c n } Cesaro converge to b, respectively c, then a ≤ bc. This implies that if in addition {b n } and {c n } are similarly ordered, then a = bc. As applications we prove that the pointwise product of two convex solid sets closed in measure is again closed in measure and a factorization result for kernels of regular integral operators on Lp-spaces.

Let Ω 1 and Ω 2 be bounded solid domains such that their associated volume potentials agree outside Ω 1 U Ω 2 . Under the assumption that one of the domains is convex, it is deduced that Ω 1 =Ω 2 .

Let a > 0 and 0 max(1, c). The reverse inequality is valid for all x, y > 0 if and only if ca < min(1, c).

The Marcinkiewicz integral Iλ (x)= ∫ (dist (y, ℝn\Ω))λ/Ω| x - y | n+λ dy, where λ > 0, plays a well-known and prominent role in harmonic analysis. In this paper, we estimate the growth of it in the limiting case λ → 0. Throughout, we assume that Ω is convex; it is interesting that this condition cannot be dropped

The purpose of this paper is to study the convergence of a path that begins at the unique fixed point of a strongly pseudo-contractive operator defined on a closed and convex subset of a reflexive Banach space and converges to a fixed point of a pseudo-contractive mapping. Primarily, it is proven that a convex combination of these two operators is indeed strongly pseudo-contractive under the weakly inward condition. This fact generalizes a result of Barbu for accretive operators.

Let D D be a convex domain in C \mathbb {C} . Let a k > 0 a_k > 0 be summable constants and let z k ∈ D z_k \in D . If the z k z_k converge sufficiently rapidly to ζ ∈ ∂ D \zeta \in \partial D from within an appropriate Stolz angle, then the function ∑ k = 1 ∞ a k / ( z − z k ) \sum _{k=1}^\infty a_k /( z - z_k ) has infinitely many zeros in D D . An example shows that the hypotheses on the z k z_k are not redundant and that two recently advanced conjectures are false.

In this short note, as a simple application of the strong result proved recently by Böhm and Wilking, we give a classification on closed manifolds with 2 2 -nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifold (with dimension ≥ 3 \ge 3 ) whose curvature operator is bounded and satisfies the pinching condition R ≥ δ tr ⁡ ( R ) 2 n ( n − 1 ) I > 0 R\ge \delta \frac {\operatorname {tr}(R)}{2n(n-1)} \mathrm {I}>0 , for some δ > 0 \delta >0 , must be compact. This provides an intrinsic analogue of a result of Hamilton on convex hypersurfaces.

Harmonic maps u from the closed disc onto bounded convex sets of the plane obey u ∈ W 1,p,p < 2.

This paper provides a version of the transversality theorem for a class of Lipschitz functions of the form f : R n × C → R n f:\textbf {R}^n\times C\to \textbf {R}^n , where C C is a convex subset of a normed vector space Z Z indexing the parameters in the problem. The set C C may be infinite-dimensional, and the notion of generic used is the measure-theoretic notion of prevalence introduced by Hunt, Sauer and Yorke (1992) and Christensen (1974). This paper also provides some results on sensitivity analysis for solutions to locally Lipschitz equations.

Every continuous function on a compact, holomorphically convex, real-analytic subset of C N can be approximated uniformly by functions holomorphic on the set.

We prove the following extended version of Simons’ inequality and present its applications. LetSSbe a set andTTbe a subset ofSS. LetCCbe a subset of a Hausdorff topological vector space which is invariant under infinite convex combinations. Letf:C×S⟶Rf: C\times S \longrightarrow \mathbb {R}be a bounded function such that the functionsf(⋅,t):C⟶Rf(\,\cdot \,,\,t):C\longrightarrow \mathbb {R}are convex for allt∈Tt \in Tandf(λx,s)=λf(x,s)f(\lambda x,\,s)=\lambda f(x,\,s)wheneverλ>0\lambda >0,x,λx∈Cx,\,\lambda x \in Cands∈S.s\in S.Let(xn)(x_n)be a sequence inCC. Assume that, for everyx∈C1={∑n=1∞λnxn:λn≥0,∑n=1∞λn=1}x \in C_1 =\left \{\sum _{n=1}^{\infty }\lambda _n\,x_n\,:\quad \lambda _n\geq 0,\,\sum _{n=1}^{\infty }\lambda _n=1\,\right \}, there existst∈Tt \in Tsatisfyingf(x,t)=sups∈Sf(x,s)f(x,\,t)=\sup _{s\in S} f(x,\,s). Then\[infx∈C1sups∈Sf(x,s)≤supt∈Tlim supnf(xn,t).\inf _{x\in C_1}\sup _{s\in S}f(x,\,s) \leq \sup _{t\in T}\limsup _{n}f(x_n,\,t).\]If−C1⊂C-C_1\subset C, then the setC1C_1in the above inequality can be replaced byconv{x1,x2,…}\textrm {conv}\{x_1, x_2, \ldots \}.

We derive the proximal normal formula for almost proximinal sets in a smooth and locally uniformly convex Banach space. Our technique leads us to show the generic Fréchet smoothness of the distance function in the case the norm is Fréchet smooth, and we derive a necessary and sufficient condition for the convexity of a Chebyshev set in a Banach space X X with norms on X X and X ∗ X^* locally uniformly convex.

First published in Proceedings of the American Mathematical Society in vol. 133, no. 6 (2005), published by the American Mathematical Society In this paper it is shown that if C and D are two closed convex subsets of a Banach space X and x ∈ C ∩D, then NC∩D(x) = NC(x)+ND(x) whenever the convex cone, (Epi σC +Epi σD), is weak* closed, where σC and NC are the support function and the normal cone of the set C respectively. This closure condition is shown to be weaker than the standard interior-pointlike conditions and the bounded linear regularity condition.

We obtain a Fourier transform estimate and an L 3/2 (R 2 )-L 3 (R 2 ) convolution estimate for certain measures on a class of convex curves in the plane.

The celebrated Gauss-Lucas theorem states that all the roots of the derivative of a complex non-constant polynomial p p lie in the convex hull of the roots of p p , called the Lucas polygon of p p . We improve the Gauss-Lucas theorem by proving that all the nontrivial roots of p ′ p’ lie in a smaller convex polygon which is obtained by a strict contraction of the Lucas polygon of p p .

It is shown that two induced norms are the same if and only if the corresponding norm numerical ranges or radii are the same, which in turn is equivalent to the vector states and mixed states arising from the norms being the same. The proofs depend on an auxiliary result of independent interest which concerns when two closed convex sets in a topological vector space are multiples of each other.

Let D be a bounded convex domain of finite type in C n with smooth boundary. In this paper, we prove the following inequality: (∫ δ0 0 M λ q (f;t)t λn(1/p-1/q-1 dt) 1/λ ≤ C p,q ∥f∥ p,0, where 1 < p < q < ∞, f ∈ H P (D), and p < A < ∞. This is a generalization of some classical result of Hardy-Littlewood for the case of the unit disc. Using this inequality, we can embed the H P space into a weighted Bergman space in a convex domain of finite type.

Within the framework of Bishop's constructive mathematics, we give conditions under which a bounded convex subset of a uniformly smooth normed space over R is located, extending results presented recently by F. Richman and H. Ishihara for subsets of a Hilbert space.

Let Omega subset of R-n be a bounded starshaped domain. In this note we consider critical points (u) over bar is an element of (ξ) over bary + W-0(1,p) (Omega; R-m) of the functional F(u, Omega) := integral(Omega) f(delu(y))dy, where f : R-m x n --> R of class C-1 satisfies the natural growth \f (xi)\ less than or equal to c(1+ \xi\(p)) for some 1less than or equal top 0, is suitably rank-one convex and in addition is strictly quasiconvex at (ξ) over bar is an element of R-m x n. We establish uniqueness results under the extra assumption that F is stationary at (u) over bar with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Muller Sverak (2003).

The composition of finitely many projections onto closed convex sets in Hilbert space arises naturally in the area of projection algorithms. We show that this composition is asymptotically regular, thus proving the so-called ``zero displacement conjecture'' from \cite{BBL}. The proof relies on a rich mix of results from monotone operator theory, fixed point theory, convex analysis, and linear algebra.

Given a precompact subset A of a type p Banach space E, where p ∈ (1, 2], we prove that for every β E [0,1) and all n ∈ N sup k 1/p' (log k) β-1 e k (aco A) ≤ c sup k 1/p' (log k) β e k (A) k≤n holds, where acoA is the absolutely convex hull of A and e k (.) denotes the k th dyadic entropy number. With this inequality we show in particular that for given A and β ∈ (-∞,1) with e n (A) ≤ n -1/p' (log n) -β for all n ∈ N the inequality e n (aco A) < c n -1/p' (log n) -β+1 holds true for all n E N. We also prove that this estimate is asymptotically optimal whenever E has no better type than p. For β = 0 this answers a question raised by Carl, Kyrezi, and Pajor which has been solved up to now only for the Hilbert space case by F. Gao.

Let X be a Banach space, C a weakly compact convex subset of X and T : C → C an asymptotically nonexpansive mapping. Under the usual assumptions on X which assure the existence of fixed point for T , we prove that the set of fixed points is a nonexpansive retract of C. We use this result to prove that all known theorems about existence of fixed point for asymptotically nonexpansive mappings can be extended to obtain a common fixed point for a commuting family of mappings. We also derive some results about convergence of iterates

Let E E be a q q -uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., ℓ p , 1 > p > ∞ \ell _p, \ 1>p>\infty ). Let T T be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset K K of E E and let ω ∈ K \omega \in K be arbitrary. Then the iteration sequence { z n } \{z_n\} defined by z 0 ∈ K , z n + 1 = ( 1 − μ n + 1 ) ω + μ n + 1 y n ; y n = ( 1 − α n ) z n + α n T z n z_0\in K, \ \ z_{n+1}=(1-\mu _{n+ 1})\omega + \mu _{n+1}y_n; \ \ y_n = (1-\alpha _n)z_n+\alpha _nTz_n , converges strongly to a fixed point of T T , provided that { μ n } \{\mu _n\} and { α n } \{\alpha _n\} have certain properties. If E E is a Hilbert space, then { z n } \{z_n\} converges strongly to the unique fixed point of T T closest to ω \omega .

We study the conjugate of the maximum, f ∨ g f \vee g , of f f and g g when f f and g g are proper convex lower semicontinuous functions on a Banach space E E . We show that ( f ∨ g ) ∗ ∗ = f ∗ ∗ ∨ g ∗ ∗ (f \vee g)^{**} = f^{**} \vee g^{**} on the bidual, E ∗ ∗ E^{**} , of E E provided that f f and g g satisfy the Attouch-Brézis constraint qualification, and we also derive formulae for ( f ∨ g ) ∗ (f \vee g)^{*} and for the “preconjugate” of f ∗ ∨ g ∗ f^{*}\vee g^{*} .

Let w: [0, oo) -p R be a continuous function with X(O) = 1 and limit ,o p(t) = 0. If t1 (I A" (0) W'(V)) is convex, then +'(t) = (I tI), t E R, is the characteristic function of an absolutely continuous probability distribution. The criterion complements P6lya's theorem and applies to characteristic functions with various types of behavior at the origin. In particular, it provides upper bounds on Kuttner's function k(A), A E (0,2), which gives the minimal value of Kc such that (1It I,) is a characteristic function. Specifically, k(5/3) < 3. Furthermore, improved lower bounds on Kuttner's function are obtained from an inequality due to Boas and Kac.

We study rigidity and regularity properties of CR maps between smooth convex hypersurfaces of finite type in C n + 1 . \mathbb C^{n+1}.

It is shown that if f f is a univalent harmonic mapping of the unit disk onto a domain having a smooth boundary arc which is convex with respect to the domain, and if the dilatation has modulus 1 on the arc, then the arc must be a line segment.

We prove a uniform boundedness theorem for families of linear operators on ordered cones. Using the concept of locally convex cones we introduce the notions of barreled cones and of weak cone-completeness. Our main result, though no straightforward generalization of the classical case, implies the Uniform Boundedness Theorem for Fréchet spaces.

L. Karp and M. Pinsky proved that, for small radius R R , the mean exit time function E R E_{R} of an extrinsic R R -ball in a hypersurface P n − 1 ⊆ R n P^{n-1} \subseteq \mathbb {R}^{n} is bounded from below by the corresponding function E ~ R \widetilde E_{R} defined on an extrinsic R R -ball in R n − 1 \mathbb {R}^{n-1} . A counterexample given by C. Mueller proves that this inequality doesn’t holds in the large. In this paper we show that, if P P is convex, then the inequality holds for all radii. Moreover, we characterize the equality and show that analogous results are true in the sphere.

The purpose of this article is to give a characterization of an upper hemicontinuous mapping with non-empty convex values being upper demicontinuous, i.e., we show that an upper hemicontinuous set-valued mapping with non-empty convex values (not necessarily compact-valued) is upper demicontinuous if and only if the set-valued mapping has no interior asymptotic plane.