1,721,006 research outputs found
Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case
No full textWe study a variational model for crack growth in elasto-plastic materials with hardening in the antiplane case. The main result is the existence of a solution to the initial value problem with prescribed time-dependent boundary conditions
Γ-Convergence and Integral Representation for a Class of Free Discontinuity Functionals
No full textWe study the Γ-limits of sequences of free discontinuity functionals with linear growth, assuming that the surface energy density is bounded. We determine the relevant properties of the Γ-limit, which lead to an integral representation result by means of integrands obtained by solving some auxiliary minimum problems on small cubes
On the pure jump nature of crack growth for a class of pressure-sensitive elasto-plastic materials
Full text linkIn the framework of a model for the quasistatic crack growth in pressure-sensitive elasto-plastic materials in the planar case, we study the properties of the length l(t) of the crack as a function of time. We prove that, under suitable technical assumptions on the crack path, the monotone function l is a pure jump function
Quasistatic crack growth in elasto-plastic materials: the two-dimensional case
No full textWe study a variational model for the quasistatic evolution of elasto-plastic materials with cracks in the case of planar small strain associative elasto-plasticity
On a notion of unilateral slope for the Mumford-Shah functional
No full textIn this paper we introduce a notion of unilateral slope for the Mumford-Shah functional, and provide an explicit formula in the case of smooth cracks. We show that the slope is not lower semicontinuous and study the corresponding relaxed functional
Decomposition results for functions with bounded variation
No full textSome decomposition results for functions with bounded variation are obtained by using Gagliardo's Theorem on the surjectivity of the trace operator from W1,1(Ω) into L1(∂Ω). More precisely, we prove that every BV function can be written as the sum of a BV function without jumps and a BV function without Cantor part. Alternatively, it can be written as the sum of a BV function without jumps and a purely ingular BV function (i.e., a function whose gradient is singular with respect to the Lebesgue measure). It can also be decomposed as the sum of a purely singular BV function and a BV function without Cantor part. We also prove similar results for the space BD of functions with bounded deformation. In particular, we show that every BD function can be written as the sum of a BD function without jumps and a BV function without Cantor part. Therefore, every BD function without Cantor part is the sum of a function whose symmetrized gradient belongs to L1 and a BV function without Cantor part
A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo"
Full text linkWe prove the existence of bounded and periodic solutions for planar systems by introducing a notion of lower and upper solutions which generalizes the classical one for scalar second order equations. The proof relies on phase plane analysis; after suitably modifying the nonlinearities, the Wazewski theory provides a solution which remains bounded in the future. For the periodic problem, the Massera Theorem applies, yielding the existence result. We then show how our result generalizes some well known theorems on the existence of bounded and of periodic solutions. Finally, we provide some corollaries on the existence of almost periodic solutions for scalar second order equations
Subharmonic Solutions of Weakly Coupled Hamiltonian Systems
Full text linkWe prove the existence of an arbitrarily large number of subharmonic solutions for a class of weakly coupled Hamiltonian systems which includes the case when the Hamiltonian function is periodic in all of its variables and its critical points are non-degenerate. Our results are obtained through a careful analysis of the dynamics of the planar components, combined with an application of a generalized version of the Poincaré–Birkhoff Theorem
Two-point boundary value problems for planar systems: A lower and upper solutions approach
Full text linkWe extend the theory of lower and upper solutions to planar systems of ordinary differential equations with separated boundary conditions, both in the well-ordered and in the non-well-ordered cases. We are able to deal with general Sturm–Liouville boundary conditions in the well-ordered case, and we analyze the Dirichlet problem in the non-well-ordered case. Our results apply in particular to scalar second order differential equations, including those driven by the mean curvature operator. Higher dimensional systems are also treated, with the same approach
Multiplicity of Periodic Solutions for Nearly Resonant Hamiltonian Systems
No full textWe prove a multiplicity result for the periodic problem associated with
a Hamiltonian system whose Hamiltonian function has a twisting part and a
nonresonant part. The possible approach to resonance together with some kind of
Landesman–Lazer conditions is also analyzed. We propose a new version of this
condition, and we also treat the so-called double resonance situation
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