We study the Dirichlet problem associated to strongly nonlinear parabolic equations involving p（x） structure in W;L;（Q）. We prove the existence of weak solutions by applying Galerkin’s approximation method.
We consider, for a bounded open domain Ω in <em>R<sup>n</sup></em> and a function <em>u</em> : Ω → <em>R<sup>m</sup></em>, the quasilinear elliptic system...We consider, for a bounded open domain Ω in <em>R<sup>n</sup></em> and a function <em>u</em> : Ω → <em>R<sup>m</sup></em>, the quasilinear elliptic system: <img src="Edit_8a3d3105-dccb-405b-bbbc-2084b80b6def.bmp" alt="" /> (1). We generalize the system (<em>QES</em>)<sub>(<em>f</em>,<em>g</em>)</sub> in considering a right hand side depending on the jacobian matrix <em>Du</em>. Here, the star in (<em>QES</em>)<sub>(<em>f</em>,<em>g</em>)</sub> indicates that <em>f </em>may depend on <em>Du</em>. In the right hand side, <em>v</em> belongs to the dual space <em>W</em><sup>-1,<em>P</em>’</sup>(Ω, <span style="white-space:nowrap;"><em>ω</em></span><sup>*</sup>,<em> R<sup>m</sup></em>), <img src="Edit_d584a286-6ceb-420c-b91f-d67f3d06d289.bmp" alt="" />, <em>f </em>and <em>g</em> satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for <em>σ</em>, but with only very mild monotonicity assumptions.展开更多
Using the theory of weighted Sobolev spaces with variable exponent and the <em>L</em><sup>1</sup>-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Di...Using the theory of weighted Sobolev spaces with variable exponent and the <em>L</em><sup>1</sup>-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Dirichlet problems generated by the Leray-Lions operator of divergence form, with right-hand side measure. Among the interest of this article is the given of a very important approach to ensure the existence of a weak solution of this type of problem and of generalization to a system with the minimum of conditions.展开更多
We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（...We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（x, u,▽u）=μ, where the right-hand side belongs to L^1（Ω）＋W^-1,p＇（x）（Ω）, -div（a（x, u,▽u）） is a Leray-Lions operator defined from W^-1,p＇（x）（Ω） into its dual and φ∈C^0（R,R^N）. The function g（x, u,▽u） is a non linear lower order term with natural growth with respect to ｜▽u｜ satisfying the sign condition, that is, g（x, u,▽u）u ≥ 0.展开更多
We study the existence of renormalized solutions for a class of nonlinear degenerated parabolic problem. The Carath6odory function satisfying the coercivity condition, the growth condition and only the large monotonic...We study the existence of renormalized solutions for a class of nonlinear degenerated parabolic problem. The Carath6odory function satisfying the coercivity condition, the growth condition and only the large monotonicity. The data belongs to LI（Q）.展开更多
The first part of this paper is devoted to study the existence of solution for nonlinear p（x） elliptic problem A（u） =u in Ω, u = 0 on Ω, with a right-hand side measure, where Ω is a bounded open set of RN, N ...The first part of this paper is devoted to study the existence of solution for nonlinear p（x） elliptic problem A（u） =u in Ω, u = 0 on Ω, with a right-hand side measure, where Ω is a bounded open set of RN, N ≥ 2 and A （u） = -div（a （x, u, u）） is a Leray-Lions operator defined from W 0 1,p（x） （Ω） in to its dual W-1,p＇（x） （Ω）. However the second part concerns the existence solution, of the following setting nonlinear elliptic problems A（u）＋g（x,u, u） = u in Ω, u = 0 on Ω. We will give some regularity results for these solutions.展开更多
文摘We study the Dirichlet problem associated to strongly nonlinear parabolic equations involving p（x） structure in W;L;（Q）. We prove the existence of weak solutions by applying Galerkin’s approximation method.
文摘We consider, for a bounded open domain Ω in <em>R<sup>n</sup></em> and a function <em>u</em> : Ω → <em>R<sup>m</sup></em>, the quasilinear elliptic system: <img src="Edit_8a3d3105-dccb-405b-bbbc-2084b80b6def.bmp" alt="" /> (1). We generalize the system (<em>QES</em>)<sub>(<em>f</em>,<em>g</em>)</sub> in considering a right hand side depending on the jacobian matrix <em>Du</em>. Here, the star in (<em>QES</em>)<sub>(<em>f</em>,<em>g</em>)</sub> indicates that <em>f </em>may depend on <em>Du</em>. In the right hand side, <em>v</em> belongs to the dual space <em>W</em><sup>-1,<em>P</em>’</sup>(Ω, <span style="white-space:nowrap;"><em>ω</em></span><sup>*</sup>,<em> R<sup>m</sup></em>), <img src="Edit_d584a286-6ceb-420c-b91f-d67f3d06d289.bmp" alt="" />, <em>f </em>and <em>g</em> satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for <em>σ</em>, but with only very mild monotonicity assumptions.
文摘Using the theory of weighted Sobolev spaces with variable exponent and the <em>L</em><sup>1</sup>-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Dirichlet problems generated by the Leray-Lions operator of divergence form, with right-hand side measure. Among the interest of this article is the given of a very important approach to ensure the existence of a weak solution of this type of problem and of generalization to a system with the minimum of conditions.
文摘We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（x, u,▽u）=μ, where the right-hand side belongs to L^1（Ω）＋W^-1,p＇（x）（Ω）, -div（a（x, u,▽u）） is a Leray-Lions operator defined from W^-1,p＇（x）（Ω） into its dual and φ∈C^0（R,R^N）. The function g（x, u,▽u） is a non linear lower order term with natural growth with respect to ｜▽u｜ satisfying the sign condition, that is, g（x, u,▽u）u ≥ 0.
文摘We study the existence of renormalized solutions for a class of nonlinear degenerated parabolic problem. The Carath6odory function satisfying the coercivity condition, the growth condition and only the large monotonicity. The data belongs to LI（Q）.
文摘The first part of this paper is devoted to study the existence of solution for nonlinear p（x） elliptic problem A（u） =u in Ω, u = 0 on Ω, with a right-hand side measure, where Ω is a bounded open set of RN, N ≥ 2 and A （u） = -div（a （x, u, u）） is a Leray-Lions operator defined from W 0 1,p（x） （Ω） in to its dual W-1,p＇（x） （Ω）. However the second part concerns the existence solution, of the following setting nonlinear elliptic problems A（u）＋g（x,u, u） = u in Ω, u = 0 on Ω. We will give some regularity results for these solutions.
