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Nonlinear Functional Analysis And Its Applications

We consider weak solutions to the Dirichlet problem $$\begin{aligned} \left\{ \begin{array}{ll} -\text {div}\,A\big (x,Du-\varTheta (u)\big )=f\quad &{}\text {in}\;\varOmega ,\\ u=0\quad &{}\text {on}\;\partial \varOmega , \end{array} \right. \end{aligned}$$
 where $$\varTheta :{\mathbb {R}}^m\rightarrow {\mathbb {M}}^{m\times n}$$
 is a continuous function assumed to satisfy a Lipschitz condition. Based on the theory of Young measures, we prove the existence result when $$f\in W^{-1,p'}(\varOmega ;{\mathbb {R}}^m)$$
 .

A weak solution to quasilinear elliptic problems with perturbed gradient

In this paper, we consider the boundary value problem of a quasilinear elliptic system in degenerate form with data belongs to the dual of Sobolev Spaces. The existence result is proved by means of Young measures and mild monotonicity assumptions.

https://journals.semnan.ac.ir/article_4194_f3af453c02ec299431f12594f9e650a1.pdf

Quasilinear elliptic systems in perturbed form

Abstract In this paper, we prove existence results in the setting of Sobolev spaces for a strongly quasilinear elliptic system by means of Young measures and mild monotonicity assumptions.

On strongly quasilinear elliptic systems with weak monotonicity

We prove an existence result for solutions of nonlinear $p$-Laplacian systems with data in generalized form:\[\left\{\begin{array}{rl}-\text{div}\,\Phi(Du-\Theta(u))&=f(x,u,Du)\quad\text{in}\;\Omega\\u&=0\quad\text{on}\;\partial\Omega\end{array}\right.\]by the theory of Young measures.

/pdf/elliptic-systems-of-p-laplacian-type-27gx3t5aqc.pdf

Elliptic Systems of $p$-Laplacian Type

Abstract We prove the existence of weak solutions to a generalized p-Laplacian systems in degenerate form. The techniques of Young measure for elliptic systems are used to prove the existence result.

Weak solutions for generalized p-Laplacian systems via Young measures

The existence of solutions to the strongly quasilinear parabolic system[frac{partial u}{partial t}-text{div},sigma(x,t,u,Du)+g(x,t,u,Du)=f,]is proved, where the source term $f$ is assumed to belong to $L^{p'}(0,T; W^{-1,p'}(Omega;R^m))$. Further, we prove the existence of a weak solution by means of the Young measures under mild monotonicity assumptions on $sigma$.

http://www.kjm-math.org/article_97170_c50bb41e07dca9346043d4702bb0d0b2.pdf

Farah Balaadich

Papers

Weak solutions for generalized p-Laplacian systems via Young measures

A weak solution to quasilinear elliptic problems with perturbed gradient

Quasilinear elliptic systems in perturbed form

On strongly quasilinear elliptic systems with weak monotonicity

Strongly Quasilinear Parabolic Systems in Divergence Form with Weak Monotonicity