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Existence Theorems for the Dirichlet Elliptic Inclusion Involving Exponential-Growth-Type Multivalued Right-Hand Side

Hôǹg Thái Nguyêñ, Dariusz Pączka (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

We present two existence results for the Dirichlet elliptic inclusion with an upper semicontinuous multivalued right-hand side in exponential-type Orlicz spaces involving a vector Laplacian, subject to Dirichlet boundary conditions on a domain Ω⊂ ℝ². The first result is obtained via the multivalued version of the Leray-Schauder principle together with the Nakano-Dieudonné sequential weak compactness criterion. The second result is obtained by using the nonsmooth variational technique together with...

Explicit Construction of Piecewise Affine Mappings with Constraints

Waldemar Pompe (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

We construct explicitly piecewise affine mappings u:ℝ ⁿ → ℝ ⁿ with affine boundary data satisfying the constraint div u = 0. As an application of the construction we give short and direct proofs of the main approximation lemmas with constraints in convex integration theory. Our approach provides direct proofs avoiding approximation by smooth mappings and works in all dimensions n ≥ 2. After a slight modification of our construction, the constraint div u = 0 can be turned into det Du = 1, giving...

Explicit formulae of distributions and densities of characteristics of a dynamic advertising and pricing model

Kurt L. Helmes, Torsten Templin (2015)

Banach Center Publications

We analyze the optimal sales process of a stochastic advertising and pricing model with constant demand elasticities. We derive explicit formulae of the densities of the (optimal) sales times and (optimal) prices when a fixed finite number of units of a product are to be sold during a finite sales period or an infinite one. Furthermore, for any time t the exact distribution of the inventory, i.e. the number of unsold items, at t is determined and will be expressed in terms of elementary functions....

Exponential convergence for a convexifying equation

Guillaume Carlier, Alfred Galichon (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.

Exponential convergence for a convexifying equation

Guillaume Carlier, Alfred Galichon (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.

Exponential convergence for a convexifying equation

Guillaume Carlier, Alfred Galichon (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.

Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks*

Zhong-Jie Han, Gen-Qi Xu (2011)

ESAIM: Control, Optimisation and Calculus of Variations


In this paper, the stability of a Timoshenko beam with time delays in the boundary input is studied. The system is fixed at the left end, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system is well-posed. By the complete spectral analysis, we show that there is a sequence of eigenvectors and generalized eigenvectors of the system operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum...

Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks*

Zhong-Jie Han, Gen-Qi Xu (2011)

ESAIM: Control, Optimisation and Calculus of Variations


In this paper, the stability of a Timoshenko beam with time delays in the boundary input is studied. The system is fixed at the left end, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system is well-posed. By the complete spectral analysis, we show that there is a sequence of eigenvectors and generalized eigenvectors of the system operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum...

Extended Hashin-Shtrikman variational principles

Petr Procházka, Jiří Šejnoha (2004)

Applications of Mathematics

Internal parameters, eigenstrains, or eigenstresses, arise in functionally graded materials, which are typically present in particulate, layered, or rock bodies. These parameters may be realized in different ways, e.g., by prestressing, temperature changes, effects of wetting, swelling, they may also represent inelastic strains, etc. In order to clarify the use of eigenparameters (eigenstrains or eigenstresses) in physical description, the classical formulation of elasticity is presented, and the...

Extending the applicability of Newton's method using nondiscrete induction

Ioannis K. Argyros, Saïd Hilout (2013)

Czechoslovak Mathematical Journal

We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete...

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