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

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 > 0 {M(x,s)\xi\cdot\xi\geq\alpha|\xi|^{2}>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 < q < 2 {1<q<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 > N 2 and \displaystyle b\in L^{\frac{2m}{2-q}}(\Omega)\text{ for some }m>\frac{N}{2}% \text{ and} (0.1) ∃ Q > 0 such that | f ( x ) | ≤ Q a ( x ) \displaystyle\exists\,Q>0\text{ such that }|f(x)|\leq Qa(x) or f ∈ L m ( Ω ) for some m > N 2 and \displaystyle f\in L^{m}(\Omega)\text{ for some }m>\frac{N}{2}\text{ and} (0.2) ∃ R > 0 such that | b ( x ) | 2 2 - q ≤ R a ( x ) . \displaystyle\exists\,R>0\text{ such that }|b(x)|^{\frac{2}{2-q}}\leq Ra(x). In addition, we also prove the existence for every f ∈ L 1 ( Ω ) {f\kern-1.0pt\in\kern-1.0ptL^{1}(\Omega)} and b ( x ) ∈ L 2 2 - q ( Ω ) {b(x)\kern-1.0pt\in\kern-1.0ptL^{\frac{2}{2-q}}(\Omega)} satisfying both conditions (0.1) and (0.2) jointly.

Abstract licence: CC BY

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