A study on the fuzzy parabolic Volterra partial integro-differential equations

This paper is concerned with aspects of the analytical fuzzy solutions of the parabolic Volterra partial integro-differential equations under generalized Hukuhara partial differentiability and it consists of two parts. The first part of this paper deals with aspects of background knowledge in fuzzy mathematics, with emphasis on the generalized Hukuhara partial differentiability. The existence and uniqueness of the solutions of the fuzzy Volterra partial integro-differential equations by considering the type of [gH - p]-differentiability of solutions are proved in this part. The second part is concerned with the central themes of this paper, using the fuzzy Laplace transform method for solving the fuzzy parabolic Volterra partial integro-differential equations with emphasis on the type of [gH - p]-differentiability of solution. We test the effectiveness of method by solving some fuzzy Volterra partial integro-differential equations of parabolic type.

A regularity result for a class of elliptic equations with lower order terms

In this paper we establish the higher differentiability of solutions to the Dirichlet problem $$\begin{aligned} {\left\{ \begin{array}{ll} \text {div} (A(x, Du)) + b(x)u(x)=f &{} \text {in}\, \Omega \\ u=0 &{} \text {on} \, \partial \Omega \end{array}\right. } \end{aligned}$$ div ( A ( x , D u ) ) + b ( x ) u ( x ) = f in Ω u = 0 on ∂ Ω under a Sobolev assumption on the partial map $$x \rightarrow A(x, \xi )$$ x → A ( x , ξ ) . The novelty here is that we take advantage from the regularizing effect of the lower order term to deal with bounded solutions.

Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions

We establish the higher differentiability of integer order of solutions to a class of obstacle problems assuming that the gradient of the obstacle possesses an extra integer differentiability property. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form\int_{\Omega}\langle\mathcal{A}(x,Du),D(\varphi-u)\rangle\,dx\geq 0\quad\text{% for all }\varphi\in\mathcal{K}_{\psi}(\Omega).The main novelty is that the operator {\mathcal{A}} satisfies the so-called {p,q}-growth conditions with p and q linked by the relation\frac{q}{p}<1+\frac{1}{n}-\frac{1}{r},for {r>n}. Here {\psi\in W^{1,p}(\Omega)} is a fixed function, called obstacle, for which we assume {D\psi\in W^{1,2q-p}_{\mathrm{loc}}(\Omega)}, and {\mathcal{K}_{\psi}=\{w\in W^{1,p}(\Omega):w\geq\psi\text{ a.e. in }\Omega\}} is the class of admissible functions. We require for the partial map {x\mapsto\mathcal{A}(x,\xi\/)} a higher differentiability of Sobolev order in the space {W^{1,r}}, with {r>n} satisfying the condition above.