1,723,135 research outputs found
Dynamics and dissipation in 2D fluid foams: The foamy continuum model
No full textThe paper establishes the governing equations of the dynamics of a foamy continuum, in the sense of Davini and Podio-Guidugli, which is expected to mimic the behavior of a monodisperse fluid foam. We model this special continuum as a dissipative ordered fluid, following the lead by Sonnet and Virga. Attention to both regular and topological transformations is taken
Piece-wise constant approximations in variational problems via W^1,p estimates
No full textPiece-wise constant approximation of first order variational problems via W −1, p estimates
C. Davini, R. Paroni
Dipartimento di Georisorse e Territorio, Via Cotonificio 114, 33100 Udine, Italy
DAP, Università di Sassari, Palazzo del Pou Salit, 07041 Alghero, Italy
A classical problem in the calculus of variations is: find the minimizers of the functional
F(v) =
W (x, v, ∇v) dx – (f , v)
Ω
among all functions v ∈ W 1, p(Ω) with trace equal to w ∈ W 1, p(Ω) over a subset (of positive length) ∂uΩ
of the boundary of Ω, where 1 < p < +∞, Ω ⊂ R2 is an open bounded set with Lipschitz boundary,
f ∈ Lq(Ω) and W : Ω × R × R2 → R is a Carathéodory function convex in the last variable and satisfying
a standard p-growth from below and above.
Different schemes have been developed in order to find an approximation of the minimizer(s) of the problem above. Probably, the most popular is the technique based on the use of continuous piece-wise affine finite elements. Higher order approximants have also been used. These on one hand give a better rate of convergence but on the other hand make the numerical scheme more complex.
Our point of view here is to consider the space which makes the numerical scheme as simple as possible, which is the space of piece-wise constant functions over triangulations of the base domain.
By using piece-wise constant functions the first problem at our hand is to define what we mean by gradient. At each nodal point xi of the triangulation of the base domain we call generalized gradient of a piece-wise constant function a suitable mean of the distributional gradient on a dual element around the point xi. This notion is then extended to the full triangulation by taking the generalized gradient constant on each dual element. Despite the given name, the generalized gradient is not a gradient, even though we show that it has some of the properties which are peculiar to a gradient. In particular we prove that if a sequence of generalized gradients weakly converges in Lp then the weak limit is a gradient. The generalized gradient instead does not have the "imbedding property" of a gradient, which is: a sequence of piece-wise constant functions which weakly converges together with the sequence of the generalized gradients does not necessarily strongly converges. This property is recovered by requiring that a certain weighted Lp norm of the jumps of the piece-wise constant function across the edges of the mesh should
tend to zero as the size of the triangulation goes to zero. This is proved by working in the W −1, p space
and using an inequality due to Necˇas. The lacking of this imbedding property strongly influences the definition of the discrete functionals which approximate the original one.
References
[1] Davini, C.; Paroni, R. External approximation of first order variational problems via W −1, p estimates. ESAIM Control Optim. Calc. Var. 14 (2008), no. 4, 802–824
Bill of Medicines for College and Nobili
No full textOne page bill of medicines for Santa Clara College and for Fr. Nobili from Dom Davini
Bill of Medicines for College and Nobili
No full textOne page bill of medicines for Santa Clara College and for Fr. Nobili from Dom Davini
Some remarks on the continuum theory of defects in solids
No full textA succinct account of the classical continuum theory of defects is given and some critical remarks on its relevance for
the mechanics of the model are discussed
Existence and uniqueness of solutions to parabolic equations with superlinear Hamiltonians
Full text linkWe give a proof of existence and uniqueness of viscosity solutions to parabolic quasi- linear equations for a fairly general class of nonconvex Hamiltonians with superlinear growth in the gradient variable. The approach is mainly based on classical techniques for uniformly parabolic quasilinear equations and on the Lipschitz estimates provided in [S. N. Armstrong and H. V. Tran, Viscosity solutions of general viscous Hamilton–Jacobi equations, Math. Ann. 361 (2015) 647–687], as well as on viscosity solution arguments
Weak KAM theory for nonregular commuting Hamiltonians
No full textIn this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax-Oleinik semigroups. This is equivalent to the solvability of an associated multi-time Hamilton-Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions and the same Aubry set, thus generalizing a result recently obtained by the second author for Tonelli Hamiltonians. We make a further step by proving that the Hamiltonians admit a common critical subsolution, strict outside their Aubry set. This subsolution can be taken of class C-1,C-1 in the Tonelli case. To prove our main results in full generality, it is crucial to establish suitable differentiability properties of the critical subsolutions on the Aubry set. These latter results are new in the purely continuous case and of independent interest
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