An evolutionary Haar-Rado type theorem
Abstract: In this paper, we study variational solutions to parabolic equations of the type $$\partial _t u - \mathrm {div}_x (D_\xi f(Du)) + D_ug(x,u) = 0$$ ∂ t u - div x ( D ξ … Show more
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Cited by 2 publications
References 27 publications
The bounded slope condition for parabolic equations with time-dependent integrands
In this paper, we study the Cauchy–Dirichlet problem $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {\text {div}} \left( D_\xi f(t, Du)\right) = 0 &{} \quad \hbox {in} \ \Omega _T, \\ u = u_o &{} \quad \hbox { on} \ \partial _{\mathcal {P}} \Omega _T,\\ \end{array} \right. \end{aligned}$$ ∂ t u - div D ξ f ( t , D u ) = 0 in Ω T , u = u o on ∂ P Ω T , where $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n is a convex and bounded domain, $$f:[0,T]\times {\mathbb {R}}^n \rightarrow {\mathbb {R}}$$ f : [ 0 , T ] × R n → R is $$L^1$$ L 1 -integrable in time and convex in the second variable. Assuming that the initial and boundary datum $$u_o:{\overline{\Omega }}\rightarrow {\mathbb {R}}$$ u o : Ω ¯ → R satisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable.
The bounded slope condition for parabolic equations with time-dependent integrands
In this paper, we study the Cauchy–Dirichlet problem $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {\text {div}} \left( D_\xi f(t, Du)\right) = 0 &{} \quad \hbox {in} \ \Omega _T, \\ u = u_o &{} \quad \hbox { on} \ \partial _{\mathcal {P}} \Omega _T,\\ \end{array} \right. \end{aligned}$$ ∂ t u - div D ξ f ( t , D u ) = 0 in Ω T , u = u o on ∂ P Ω T , where $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n is a convex and bounded domain, $$f:[0,T]\times {\mathbb {R}}^n \rightarrow {\mathbb {R}}$$ f : [ 0 , T ] × R n → R is $$L^1$$ L 1 -integrable in time and convex in the second variable. Assuming that the initial and boundary datum $$u_o:{\overline{\Omega }}\rightarrow {\mathbb {R}}$$ u o : Ω ¯ → R satisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable.
We study regularity properties of variational solutions to a class of Cauchy–Dirichlet problems of the form { ∂ t u - div x ( D ξ f ( D u ) ) = 0 in Ω T , u = u 0 on ∂ 𝒫 Ω T . \left\{\begin{aligned} \displaystyle\partial_{t}u-\operatorname{div}_{x}(D_{% \xi}f(Du))&\displaystyle=0&&\displaystyle\phantom{}\text{in }\Omega_{T},\\ \displaystyle u&\displaystyle=u_{0}&&\displaystyle\phantom{}\text{on }\partial% _{\mathcal{P}}\Omega_{T}.\end{aligned}\right. We do not impose any growth conditions from above on f : ℝ n → ℝ {f\colon\mathbb{R}^{n}\to\mathbb{R}} , but only require it to be convex and coercive. The domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is mainly supposed to be bounded and convex, and for the time-independent boundary datum u 0 : Ω ¯ → ℝ {u_{0}\colon\overline{\Omega}\to\mathbb{R}} we only require continuity. These requirements are weaker than a one-sided bounded slope condition. We prove global continuity of the unique variational solution u : Ω T → ℝ {u\colon\Omega_{T}\to\mathbb{R}} . If the boundary datum is Lipschitz continuous, we obtain global Hölder continuity of the solution.
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