Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space

By Piotr Gwiazda, Iwona Skrzypczak, Anna Zatorska-Goldstein. 2017

80 citations recorded by the source corpus.

Abstract We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak–Orlicz space avoiding growth restrictions. Namely, we consider − div A ( x , ∇ u ) = f ∈ L 1 ( Ω ) , on a Lipschitz bounded domain in R N . The growth of the monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N-function M. The approach does not require any particular type of growth condition of M or its conjugate M ⁎ (neither Δ 2 , nor ∇ 2 ). The condition we impose is log-Holder continuity of M, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d45c28f72eaf04d23d1e3993d6a565b002fa6785

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In Applied Mathematics

1. Iwona Chlebicka
2. Ferenc Weisz
3. Dachun Yang
4. Yong Jiao
5. Guangheng Xie
6. Youssef Ahmida
7. Piotr Gwiazda
8. A. Youssfi

In Computational Theory and Mathematics

1. Iwona Chlebicka
2. Youssef Ahmida
3. Piotr Gwiazda
4. Dachun Yang
5. Ferenc Weisz
6. Yong Jiao
7. A. Youssfi
8. Paolo Marcellini

In Mathematical Physics

1. Iwona Chlebicka
2. Ferenc Weisz
3. Guangheng Xie
4. Dachun Yang
5. Yong Jiao
6. Piotr Gwiazda
7. Wen Yuan
8. Yuan Zhou

Citations in the corpus

• Gossez's approximation theorems in Musielak–Orlicz–Sobolev spaces (2017, background): “[18, 20]) to provide a modular density result not for W 0 LM (Ω), but for V m 0 LM (Ω) = {u ∈ W 1,1 0 (Ω) ∩W (Ω) : Du ∈ LM (Ω)}, where W 1,1 0 (Ω) is defined as a closure of C ∞ 0 (Ω) in W (1,1)(Ω)–norm.”
• A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces (2018, methodology): “Renormalized solutions to L(1)-data problems in nonreflexive anisotropic Musielak-Orlicz spaces are considered in the elliptic setting in [106, 112, 113] and in the parabolic one in [47, 48, 114].”
• New martingale inequalities and applications to Fourier analysis (2018, background): “The Musielak–Orlicz spaces not only have their own interest, but they are also very useful in partial differential equations [2, 5, 28, 26], in calculus of variations [13], in image restoration [27, 43] and in fluid dynamics [73, 51].”