Regularizing Effect of Two Hypotheses on the Interplay Between Coefficients in Some Hamilton–Jacobi Equations

Abstract

We study of the regularizing effect of the interaction between the coefficient of the zero-order term and the lower-order term in quasilinear Dirichlet problems whose model is ∫ Ω M ⁢ ( x , u ) ⁢ ∇ ⁡ u ⋅ ∇ ⁡ φ + ∫ Ω a ⁢ ( x ) ⁢ u ⁢ φ = ∫ Ω b ⁢ ( x ) ⁢ | ∇ ⁡ u | q ⁢ φ + ∫ Ω f ⁢ ( x ) ⁢ φ for all ⁢ φ ∈ W 0 1 , 2 ⁢ ( Ω ) ∩ L ∞ ⁢ ( Ω ) , \int_{\Omega}M(x,u)\nabla u\cdot\nabla\varphi+\int_{\Omega}a(x)u\varphi=\int_{% \Omega}b(x)|\nabla u|^{q}\varphi+\int_{\Omega}f(x)\varphi\quad\text{for all }% \varphi\in W_{0}^{1,2}(\Omega)\cap L^{\infty}(\Omega), where Ω is a bounded open set of ℝ N {\mathbb{R}^{N}} , M ⁢ ( x , s ) {M(x,s)} is a Carathéodory matrix on Ω × ℝ {\Omega\times\mathbb{R}} which is elliptic (that is, M ⁢ ( x , s ) ⁢ ξ ⋅ ξ ≥ α ⁢ | ξ | 2 &gt; 0 {M(x,s)\xi\cdot\xi\geq\alpha|\xi|^{2}&gt;0} for every ( x , s , ξ ) ∈ Ω × ℝ × ( ℝ N ∖ { 0 } ) {(x,s,\xi)\in\Omega\times\mathbb{R}\times(\mathbb{R}^{N}\setminus\{0\})} ) and bounded (that is, | M ⁢ ( x , s ) | ≤ β {|M(x,s)|\leq\beta} for every ( x , s ) ∈ Ω × ℝ {(x,s)\in\Omega\times\mathbb{R}} ), b ⁢ ( x ) ∈ L 2 2 - q ⁢ ( Ω ) {b(x)\in L^{\frac{2}{2-q}}(\Omega)} , 1 &lt; q &lt; 2 {1&lt;q&lt;2} and 0 ≤ a ⁢ ( x ) ∈ L 1 ⁢ ( Ω ) {0\leq a(x)\in L^{1}(\Omega)} . We prove the existence of a weak solution u belonging to W 0 1 , 2 ⁢ ( Ω ) {W_{0}^{1,2}(\Omega)} and to L ∞ ⁢ ( Ω ) {L^{\infty}(\Omega)} when either b ∈ L 2 ⁢ m 2 - q ⁢ ( Ω ) ⁢ for some ⁢ m &gt; N 2 ⁢ and \displaystyle b\in L^{\frac{2m}{2-q}}(\Omega)\text{ for some }m&gt;\frac{N}{2}% \text{ and} (0.1) ∃ Q &gt; 0 ⁢ such that ⁢ | f ⁢ ( x ) | ≤ Q ⁢ a ⁢ ( x ) <jats:graphic xmlns