Partial regularity for Double Phase obstacle problems
We investigate partial regularity for vector valued local minimizers of double phase functionals, under vectorial obstacle type constraints satisfying appropriate topological properties.
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arXiv subjects
Cintia Pacchiano Camacho
Publications and source records attributed to Cintia Pacchiano Camacho.
Local Boundedness of Local Minimizers for a Class of Nonlinear Elliptic Systems with General Growth
In this paper, we prove the local boundedness of solutions to systems of partial differential equations in divergence form. More specifically, we consider systems that include the first variations of functionals depending on the spatial variable and exhibiting nonstandard growth with respect to the gradient, such as $$\int_Ω \left( 1+ h(|Du|)\right) ^{α(x)} \, d x,$$ where the convex function $h=h(t)$ does not satisfy the so-called $Δ_2$ property and does not exhibit the conventional polynomial growth behavior.
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On the Lavrentiev gap for manifold-valued maps
We investigate the validity and the failure of modular density of smooth maps on compact manifolds.
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Boundary regularity for quasiminima of double-phase problems on metric spaces
We give a sufficient condition for Hölder continuity at a boundary point for quasiminima of double-phase functionals of $p,q$-Laplace type, in the setting of metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. We use a variational approach based on De Giorgi-type conditions to give a pointwise estimate near a boundary point. The proofs are based on a careful phase analysis and estimates in the intrinsic geometries.
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Unified a-priori estimates for minimizers under $p,q-$growth and exponential growth
We propose some general growth conditions on the function $% f=f\left( x,ξ\right) $, including the so-called natural growth, or polynomial, or $p,q-$growth conditions, or even exponential growth, in order to obtain that any local minimizer of the energy integral $\;\int_{Ω}f\left( x,Du\right) dx\,$ is locally Lipschitz continuous in $Ω$. In fact this is the fundamental step for further regularity: the local boundedness of the gradient of any Lipschitz continuous local minimizer a-posteriori makes irrelevant the behavior of the integrand $f\left( x,ξ\right) $ as $\left\vert ξ\right\vert \rightarrow +\infty $; i.e., the general growth conditions a posteriori are reduced to a standard growth, with the possibility to apply the classical regularity theory. In other words, we reduce some classes of \textit{non-uniform} elliptic variational problems to a context of uniform ellipticity.
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Regularity results for quasiminima of a class of double phase problems
We prove boundedness, Hölder continuity, Harnack inequality results for local quasiminima to elliptic double phase problems of $p$-Laplace type in the general context of metric measure spaces. The proofs follow a variational approach and they are based on the De Giorgi method, a careful phase analysis and estimates in the intrinsic geometries.
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Gradient higher integrability for double phase problems on metric measure spaces
We study local and global higher integrability properties for quasiminimizers of a class of double-phase integrals characterized by nonstandard growth conditions. We work purely on a variational level in the setting of a metric measure space with a doubling measure and a Poincaré inequality. The main novelty is an intrinsic approach to double-phase Sobolev-Poincaré inequalities.
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Higher integrability and stability of $(p,q)$-quasiminimizers
Using purely variational methods, we prove local and global higher integrability results for upper gradients of quasiminimizers of a $(p,q)$-Dirichlet integral with fixed boundary data, assuming it belongs to a slightly better Newtonian space. We also obtain a stability property with respect to the varying exponents $p$ and $q$. The setting is a doubling metric measure space supporting a Poincaré inequality.
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Variational solutions to the total variation flow on metric measure spaces
We discuss a purely variational approach to the total variation flow on metric measure spaces with a doubling measure and a Poincaré inequality. We apply the concept of parabolic De Giorgi classes together with upper gradients, Newtonian spaces and functions of bounded variation to prove a necessary and sufficient condition for a variational solution to be continuous at a given point.
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Regularity properties for quasiminimizers of a $(p,q)$-Dirichlet integral
Using a variational approach we study interior regularity for quasiminimizers of a $(p,q)$-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincaré inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality, the strong maximum principle, and Liouville's Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider $(p,q)$-minimizers and we give an estimate for their oscillation at boundary points.
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Existence of parabolic minimizers to the total variation flow on metric measure spaces
We give an existence proof for variational solutions $u$ associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $(\mathcal{X}, d, μ)$ satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy-Dirichlet datum $u_0$ on the parabolic boundary of a space-time-cylinder $Ω\times (0, T)$ with $Ω\subset \mathcal{X}$ an open set and $T > 0$, we prove existence in the weak parabolic function space $L^1_w(0, T; \mathrm{BV}(Ω))$. In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $\mathrm{BV}$-valued parabolic function spaces. We argue completely on a variational level.
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