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The doubly degenerate nutrient taxis model u t = ∇ ⋅ ( u v ∇ u ) − ∇ ⋅ ( u 2 v ∇ v ) + l u v , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , is considered in smoothly bounded convex subdomains of the plane, with l ≥ 0 . It is shown that for any p > 2 and each fixed nonnegative u 0 ∈ W 1 , ∞ ( Ω ) , a smallness condition exclusively involving v 0 can be identified as sufficient to ensure that an associated no-flux type initial–boundary value problem with ( u , v ) | t = 0 = ( u 0 , v 0 ) admits a global weak solution satisfying ess sup t > 0 ‖ u ( ⋅ , t ) ‖ L p ( Ω ) ∞ . The proof relies on the use of an apparently novel class of functional inequalities which provide estimates from below for certain Dirichlet integrals involving possibly degenerate weight functions.

We give a simple necessary and sufficient condition for the existence of star supremum for two arbitrary operators on a Hilbert space. It is shown that the results of Hartwig (1979) [15] , and Janowitz (1983) [18] , when applied to the ring of bounded operators on a Hilbert space, require much simpler conditions. We also give a complete answer to a question stated by Hartwig and Drazin (1982) in [16] , regarding the extremal values of range and null-space of the star infimum.

We consider a supremal functional of the form F ( u ) = ess sup x ∈ Ω f ( x , D u ( x ) ) where Ω ⊆ R N is a regular bounded open set, u ∈ W 1 , ∞ ( Ω ) and f is a Borel function. Assuming that the intrinsic distances d F λ ( x , y ) ≔ sup { u ( x ) − u ( y ) : F ( u ) ≤ λ } are locally equivalent to the euclidean one for every λ > inf W 1 , ∞ ( Ω ) F , we give a description of the sublevel sets of the weak ∗ -lower semicontinuous envelope of F in terms of the sub-level sets of the difference quotient functionals R d F λ ( u ) ≔ sup x ≠ y u ( x ) − u ( y ) d F λ ( x , y ) . As a consequence we prove that the relaxed functional of positive 1-homogeneous supremal functionals coincides with R d F 1 . Moreover, for a more general supremal functional F (a priori non coercive), we prove that the sublevel sets of its relaxed functionals with respect to the weak ∗ topology, the weak ∗ convergence and the uniform convergence are convex. The proof of these results relies both on a deep analysis of the intrinsic distances associated to F and on a careful use of variational tools such as Γ -convergence.

This paper first studies the measure theoretic pressure of measures that are not necessarily ergodic. We define the measure theoretic pressure of an invariant measure (not necessarily ergodic) via the Carath\'{e}odory-Pesin structure described in \cite{Pes97}, and show that this quantity is equal to the essential supremum of the free energy of the measures in an ergodic decomposition. To the best of our knowledge, this formula is new even for entropy. Meanwhile, we define the measure theoretic pressure in another way by using separated sets, it is showed that this quantity is exactly the free energy if the measure is ergodic. Particularly, if the dynamical system satisfies the uniform separation condition and the ergodic measures are entropy dense, this quantity is still equal to the the free energy even if the measure is non-ergodic. As an application of the main result, we find that the Hausdorff dimension of an invariant measure supported on an average conformal repeller is given by the zero of the measure theoretic pressure of this measure. Furthermore, if a hyperbolic diffeomorphism is average conformal and volume-preserving, the Hausdorff dimension of any invariant measure on the hyperbolic set is equal to the sum of the zeros of measure theoretic pressure restricted to stable and unstable directions., Comment: 23 pages

In this work we prove continuity of solutions with respect to initial conditions and couple parameters and we prove joint upper semicontinuity of a family of global attractors for the problem \begin{eqnarray} &\frac{\partial u_{s}}{\partial t}(t)-\textrm{div}(D_s|\nabla u_{s}|^{p_s(x)-2}\nabla u_{s})+|u_s|^{p_s(x)-2}u_s=B(u_{s}(t)),\;\; t>0,\\ &u_{s}(0)=u_{0s}, \end{eqnarray} under homogeneous Neumann boundary conditions, $u_{0s}\in H:=L^2(\Omega),$ $\Omega\subset\mathbb{R}^n$ ($n\geq 1$) is a smooth bounded domain, $B:H\rightarrow H$ is a globally Lipschitz map with Lipschitz constant $L\geq 0$, $D_s\in[1,\infty)$, $p_s(\cdot)\in C(\bar{\Omega})$, $p_s^-:=\textrm{ess inf}\;p_s\geq p,$ $p_s^+:=\textrm{ess sup}\;p_s\leq a,$ for all $s\in \mathbb{N},$ when $p_s(\cdot)\rightarrow p$ in $L^\infty(\Omega)$ and $D_s\rightarrow\infty$ as $s\rightarrow\infty,$ with $a,p>2$ positive constants.

In this paper we show that if the supremal functional $$F(V,B)= \mathop{\rm ess sup} \limits_{x \in B} f(x,DV (x))$$ is sequentially weak* lower semicontinuous on $${W^{1,\infty}(B, \mathbb{R}^d)}$$ for every open set $${B \subseteq \Omega}$$ (where $${\Omega}$$ is a fixed open set of $${\mathbb{R}^N}$$ ), then $${f(x,\cdot)}$$ is rank-one level convex for a.e $${x \in \Omega}$$ . Next, we provide an example of a weak Morrey quasiconvex function which is not strong Morrey quasiconvex. Finally we discuss the L p -approximation of a supremal functional F via $${\Gamma}$$ -convergence when f is a non-negative and coercive Caratheodory function.

This paper deals with the following variational problem with variable exponents { − div ( ( 1 + | ∇ u | p ( x ) 1 + | ∇ u | 2 p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) , in Ω , u = 0 , on ∂ Ω , where Ω ⊆ R N ( N ≥ 2 ) be a bounded domain with the smooth boundary ∂Ω, p : Ω ‾ → R is a Lipschitz continuous function with 1 p − : = ess inf x ∈ Ω ⁡ p ( x ) ≤ p ( x ) ≤ p + : = ess sup x ∈ Ω ⁡ p ( x ) N and f ∈ C ( Ω × R , R ) is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz type condition. Under three different superlinear conditions on f at infinity, we prove that the above equation has at least a nontrivial solution. Moreover, the existence of infinite many solutions is proved for odd nonlinearity. Especially, some new tricks are introduced to show the boundedness of Cerami sequences.

Consider in a real Hilbert space H the differential equation (E) : p ( t ) u ′ ′ ( t ) + q ( t ) u ′ ( t ) ∈ A u ( t ) + f ( t ) , for a.a. t ∈ R + = [ 0 , ∞ ) , with the condition u ( 0 ) = x ∈ D ( A ) ¯ , where A : D ( A ) ⊂ H → H is a (possibly set-valued) maximal monotone operator, with [ 0 , 0 ] ∈ A (or, more generally, 0 ∈ R ( A ) ); p , q ∈ L ∞ ( R + ) , with ess inf p > 0 and either ess inf q > 0 or ess sup q 0 . Recall that equation (E) in the case p ≡ 1 , q ≡ 0 , f ≡ 0 , subject to u ( 0 ) = x and sup t ≥ 0 ‖ u ( t ) ‖ ∞ , was investigated in the early 1970s by V. Barbu, who derived in particular from his results a definition for the square root of the nonlinear operator A . Subsequently H. Brezis, N.H. Pavel, L. Veron and others have paid attention to equation (E) . In this paper we prove the existence and uniqueness of the solution to equation (E) subject to u ( 0 ) = x ∈ D ( A ) in the weighted space X = L b 2 ( R + ; H ) , where b ( t ) = a ( t ) / p ( t ) , a ( t ) = exp ( ∫ 0 t q ( s ) / p ( s ) d s ) , under our weak assumptions on p and q (see above) and f ∈ X . For x ∈ D ( A ) ¯ we prove the existence of a generalized solution. This is a classic solution if p ≡ 1 , q ≡ c ∈ R ∖ { 0 } . If p ≡ 1 , q ( t ) ≡ c ∈ R ∖ { 0 } , f ≡ 0 the solutions give rise to a nonlinear semigroup of contractions. If A is linear its infinitesimal generator G is given by G = − ( c / 2 ) I − ( c 2 / 4 ) I + A .

The present article is devoted to the study of a constrained weighted total variation minimization problem, which may be viewed as a relaxation of a generalized Cheeger problem and is motivated by landslide modeling. Using the fact that the set of minimizers is invariant by a wide class of monotone transformations, we prove that level sets of minimizers are generalized Cheeger sets and obtain qualitative properties of the minimizers: they are all bounded and all achieve their essential supremum on a set of positive measure.

For conformal hyperbolic flows, we establish explicit formulas for the Hausdorff dimension and for the pointwise dimension of an arbitrary invariant measure. We emphasize that these measures are not necessarily ergodic. The formula for the pointwise dimension is expressed in terms of the local entropy and of the Lyapunov exponents. We note that this formula was obtained before only in the special case of (ergodic) equilibrium measures, and these always possess a local product structure (which is not the case for arbitrary invariant measures). The formula for the pointwise dimension allows us to show that the Hausdorff dimension of a (nonergodic) invariant measure is equal to the essential supremum of the Hausdorff dimension of the measures in an ergodic decomposition.

Sufficient conditions are obtained so that a weak subsolution of a class of quasilinear degenerate parabolic equations, bounded from above on theparabolic boundary of the cylinder Q, turns out to be bounded from above in Q.

We study the lower semicontinuity properties and existence of a minimizer of the functional F( u)= ess sup x∈� f (x,u(x),Du(x)) on W 1,∞ (� ; R m ). We introduce the notions of Morrey quasiconvexity, polyquasiconvexity, and rank-one quasiconvexity, all stemming from the notion of quasiconvexity (= convex level sets) of f in the last variable. We also formally derive the Aronsson-Euler equation for such problems.  2001 Editions scientifiques et medicales Elsevier SAS

We consider a problem of minimization of convex function f ( x ) over the convex region R where the objective function and the feasible region have a common direction of recession. In cases when one of these directions is not in the constancy space of the objective function, then the minimal solution is not achieved even if the function f ( x ) is bounded below over the region R . Many algorithms, if applied to this class of programmes, do not guarantee convergence to the global infimum. Our approach to this problem leads to derivation of the equation of the feasible parametrized curve C ( t ), such that the infimum of the logarithmic penalty function along this curve is equal to the global infimum of the objective function over the region R . We show that if all functions defining the program are analytic, then C ( t ) is also an analytic function. The equation of the curve can be successfully used to determine the global infimum (in particular, unboundedness) of the convex constrained programmes in cases when the application of classical methods, such as the steepest descent method, fails to converge to the global infimum.

We give some sufficient conditions of deterioration of smoothness under the operation of convolution. We show that the convolution of two probability densities which are restrictions to R of entire functions can possess infinite essential supremum on each interval. © 1998 Academic Press.

Let $$h(t,x): = p.v. \sum\limits_{n \in Z\backslash \left| 0 \right|} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} = \mathop {\lim }\limits_{N \to \infty } \sum\limits_{0< \left| n \right| \leqslant N} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} $$ (\((i = \sqrt { - 1;} t,x\)-real variables). It is proved that in the rectangle\(D: = \left\{ {(t,x):0< t< 1,\left| x \right| \leqslant \frac{1}{2}} \right\}\), the function h satisfies the followingfunctional inequality: $$\left| {h(t,x)} \right| \leqslant \sqrt t \left| {h\left( {\frac{1}{t},\frac{x}{t}} \right)} \right| + c,$$ where c is an absolute positive constant. Iterations of this relation provide another, more elementary, proof of the known global boundedness result $$\left\| {h; L^\infty (E^2 )} \right\| : = ess sup \left| {h(t,x)} \right|< \infty .$$ The above functional inequality is derived from a general duality relation, of theta-function type, for solutions of the Cauchy initial value problem for Schrodinger equation of a free particle.

In this paper, we prove that the bounded set of fuzzy numbers must exist supremum and infimum and give the concrete representation of supremum and infimum. As an application, we obtain that the continuous fuzzy-valued function on a closed interval exists supremum and infimum and give the precise representation. We also show that the bounded fuzzy-valued function on a closed interval can define the lower and upper sums and the lower and upper integrals of Riemann and Riemann–Stieltjes by the usual way.

Let D ≌ RN , 0 0}, where ess sup ƒ denotes the essential supremum of ƒ. These properties can be used for computing ess sup ƒ. As example, two algorithms are stated. If the function ƒ is dense, or lower semicontinuous, or if −ƒ is robust, then sup ƒ = ess sup ƒ. In this case, the algorithms mentioned can be applied for determining the supremum of ƒ, i.e., also the global maximum of ƒ if it exists.

We discuss existence and multiplicity of solutions of the periodic problem for the curvature-like equation\begin{equation*}-\Big( u'/{ \sqrt{a^2+{u'}^2}}\Big)' = f(t,u) \end{equation*} by means of variational techniques in the space of bounded variation functions. As $a= 0$ is allowed,both the prescribed curvature equation and the $1$-Laplace equation are considered. We are concerned with the case where the right-hand side $f$ of the equation interacts with the beginning of the spectrum of the $1$-Laplace operator with periodic boundary conditions on $[0,T]$, being mainly interested in the situation where $\supess{[0,T]\times\RR}{f(t,s)} $ may differ from $-\infess{[0,T]\times \RR}{f(t,s)}$.

Fixed a bounded open set Ω of R , we completely characterize the weak* lower semicontinuity of functionals of the form F (u,A) = ess sup x∈A f(x, u(x), Du(x)) defined for every u ∈ W 1,∞(Ω) and for every open subset A ⊂ Ω. Without a continuity assumption on f(·, u, ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it can be represented through a level convex function. Then we study the properties of the lower semicontinuous envelope F of F . A complete relaxation theorem is shown in the case where f is a continuous function. In the case f = f(x, ξ) is only a Caratheodory function, we show that F coincides with the level convex envelope of F . Mathematics Subject Classification (2000): 47J20, 58B20, 49J45.

The subject of this paper is the Pontryagin maximum principle for the minimax problem of optimal control. That is, we will derive the necessary conditions which a control must satisfy in order to be optimal for the problem in which the cost is measured pointwise in time by the essential supremum over t of some function of time, the state, and the control. It is desired to use controls to minimize this cost

Lower bounds of essential supremum of solutions of quasilinear elliptic systems are obtained, using the method of control of essential supremum published in the paper by Korkut, Pasic and Žubrinic for the scalar case, [J. Differential Equations 170 (2001) 240-280] (the method was introduced by the second author). Results about generating singularities and lower $L^p$-bounds of the gradient of solutions of quasilinear systems are derived.

We study quasilinear elliptic equations of Leray–Lions type in W 1, p ( Ω ), maximum principles, nonexistence and existence of solutions, the control of lower (upper) bound for essential supremum (essential infimum) of solutions, sign-changing solutions, local and global oscillation of solutions, geometry of domain, generating singularities of solutions, and lower bounds on constants appearing in Schauder, Agmon, Douglis, and Nirenberg estimates.

We consider a constrained maximization problem with a linear fractional function f over any closed and unbounded set X. Starting the analysis from the behavior of unbounded sequences in X, we find necessary and sufficient conditions for the supremum to be finite. We point out that when f does not have maximum value, the supremum is not necessarily attained along a recession direction. It is known that this fundamental property holds in the classical fractional problem with a polyhedral feasible region and we give a new proof of this result by using our approach and the Representation Theorem.

Let (S, d) be a Polish space, M (S) the set of all probability measures on the σ-field of Borel subsets of S, and ⋁p = ⋁p(S) the set of all Pψ⋁(S) with ∝ d(o, ·)p dP D p (P,Q); = inf ∫d p dγ q γ ∈⋁(S 2 ) has marginal P,Q , if p ∞ inf{λ− ess sup d∥λψ⋁(S 2 has marginals P,Q}, ∞ where q:= min {1, 1 p )} . It is known, at least for 0

In this paper we consider linear systems which are stable and examine the robustness of this property. The perturbations are assumed to have unbounded structure and we determine the smallest perturbation which destroys stability. We begin with an abstract analysis of the problem, but in later sections specialize to three types of systems: those whose mild solutions are given in terms of 1. (a) strongly continuous semigroups, 2. (b) resolvent operators-integrodifferential systems, 3. (c) evolution operators-time varying systems. For B a Banach space, L2(t0, T; B) denotes the space of all square integrable functions defined on [t0, T] with values in B. For B1, B2 Banach spaces, [t0, T] denotes the space of strongly measurable functions f(·):[t0, T]→L(B1, B2) with ess sup ∥f(·)∥⩽∞.B−1(t0, ∞;L(B1, B2)) denotes the space of strongly measurable functions f(·): [t0, ∞) → L(B1, B2) with ∥f(t)∥ 0 and k(·) ϵ L1(t0, ∞).

The maximum amplitude cost of a control function u(t) taken to be ess sup g(t, u(t)), where g(t, u) is a given function. (A particular example is g(t, u) = the norm of u.) We consider linear systems with this cost functional. The existence of optimal control is proved, and it is shown that the ess sup is uniformly essential with respect to the optimal controls. Properties of the extended attainable set are discussed and compared with the case of an integral cost. Finally, we show in what sense a cost functional of the form (∝ g(t, u(t)) q ) 1 q approximates the ess sup cost functional.

Let Z and R be the set of all integers and real numbers, respectively, and let Q = [0,27c] x [0, 2711. Let L’(Q) be the space of measurable real-valued functions U: Sz + R which are Lebesgue integrable over 52 with usual norm II.jI L1. Let L2(Q) be the space of measurable real-valued functions U: Sz + R which are Lebesgue square integrable over Q with usual inner product ( , ) and usual norm I/ . ljL2 and let L”(Q) be the space of measurable real-valued functions U: Q + R which are essentially bounded with the norm 1) u 11 Lm = ess sup 1 u( t, x) I . (I. x) E R Let C“(Q) be the space of all continuous functions u.: 52 --f R such that the partial derivatives up to order k with respect to both variables are continuous on 52, while C(Q) is used for Co(Q) with the usual norm 11. Iloo.

This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard in the space Xcp of functions f on R + =(0,∞) such that ∫ 0 ∞ u c f(u) p du u ess sup u>0 u c |f(u)| for c∈ R =(−∞,∞) , in particular in the space Lp(0,∞) (1⩽p⩽∞). The semigroup property and its generalizations are established for the generalized Hadamard-type fractional integration operators under consideration. Conditions are also given for the boundedness in Xcp of these operators; they involve Kummer confluent hypergeometric functions as kernels.

We study variational problems involving nonlocal supremal functionals $L^\infty(\Omega;\mathbb{R}^m) \ni u\mapsto {\rm ess sup}_{(x,y)\in \Omega\times \Omega} W(u(x), u(y)),$ where $\Omega\subset \mathbb{R}^n$ is a bounded, open set and $W:\mathbb{R}^m\times\mathbb{R}^m\to \mathbb{R}$ is a suitable function. Motivated by existence theory via the direct method, we identify a necessary and sufficient condition for $L^\infty$-weak$^\ast$ lower semicontinuity of these functionals, namely, separate level convexity of a symmetrized and suitably diagonalized version of the supremands. More generally, we show that the supremal structure of the functionals is preserved during the process of relaxation. Whether the same statement holds in the related context of double-integral functionals is currently still open. Our proof relies substantially on the connection between supremal and indicator functionals. This allows us to recast the relaxation problem into characterizing weak$^\ast$ closures of a class of nonlocal inclusions, which is of independent interest. To illustrate the theory, we determine explicit relaxation formulas for examples of functionals with different multi-well supremands., Comment: 30 pages