Anisotropic parabolic problems with measures data

Mokhtari, Fares

doi:10.7153/dea-02-09

Cited by 10 publications

(21 citation statements)

References 13 publications

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“…This Lipschitz function satisfies T ϵ (0) = 0,| T ϵ ( σ )|≤ ϵ , and

T_{ϵ}^{'} (σ) = \{\begin{matrix} 1, & | σ | \leq ϵ; \\ 0, & | σ | > ϵ . \end{matrix}

By the definitions of μ ( x ) and E 4 , we can write

\int_{E_{4}} μ (x) dx \leq \int_{E_{4}} (\overset{a}{false} (x, D u_{n}) - \overset{a}{false} (x, D u_{m})) D T_{ϵ} (u_{n} - u_{m}) dx .

Let θ be a cutoff function as in . Using for all m , n , T ϵ ( u n − u m ) θ as test function in for ( u n ) and ( u m ) and then subtracting the results, by , we obtain

\begin{matrix} \int_{B_{r}} (\hat{a} (x, D u_{n}) - \hat{a} (x, D u_{m})) D T_{ϵ} (u_{n} - u_{m}) dx \leq C_{10} ϵ (1 + ∥ f ∥_{L^{1} (B_{2 r})} + \sum_{i = 1}^{N} \int_{B_{2 r}} (| a_{i} (x, D u_{n}) | + | a_{i} (x, D u_{m}) |) dx) . \end{matrix}

As in , in order to estimate the last term in , let

…”

Section: Statements Of Resultsmentioning

confidence: 99%

“…in [5] and approach p i ../ D 1 without getting out of the standard functional framework. Indeed, we may replace it by (17)-(18) which is better than (20) if p../ < N and .R N // 0 , so that the result of this paper deals with irregular data.…”

Section: Statements Of Resultsmentioning

confidence: 99%

“…As in [20], in order to estimate the last term in (55), let r i ../ : R N ! .1, C1/ be a continuous functions such that…”

Section: Proof Of Lemma 35mentioning

confidence: 99%

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Nonlinear anisotropic elliptic equations in with variable exponents and locally integrable data

Mokhtari

2016

Math Methods in App Sciences

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In this paper, we prove existence and regularity results for weak solutions in the framework of anisotropic Sobolev spaces for a class of nonlinear anisotropic elliptic equations in the whole double-struckRN with variable exponents and locally integrable data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results. The functional setting involves Lebesgue–Sobolev spaces with variable exponents. Copyright © 2016 John Wiley & Sons, Ltd.

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“…This Lipschitz function satisfies T ϵ (0) = 0,| T ϵ ( σ )|≤ ϵ , and

T_{ϵ}^{'} (σ) = \{\begin{matrix} 1, & | σ | \leq ϵ; \\ 0, & | σ | > ϵ . \end{matrix}

By the definitions of μ ( x ) and E 4 , we can write

\int_{E_{4}} μ (x) dx \leq \int_{E_{4}} (\overset{a}{false} (x, D u_{n}) - \overset{a}{false} (x, D u_{m})) D T_{ϵ} (u_{n} - u_{m}) dx .

Let θ be a cutoff function as in . Using for all m , n , T ϵ ( u n − u m ) θ as test function in for ( u n ) and ( u m ) and then subtracting the results, by , we obtain

\begin{matrix} \int_{B_{r}} (\hat{a} (x, D u_{n}) - \hat{a} (x, D u_{m})) D T_{ϵ} (u_{n} - u_{m}) dx \leq C_{10} ϵ (1 + ∥ f ∥_{L^{1} (B_{2 r})} + \sum_{i = 1}^{N} \int_{B_{2 r}} (| a_{i} (x, D u_{n}) | + | a_{i} (x, D u_{m}) |) dx) . \end{matrix}

As in , in order to estimate the last term in , let

…”

Section: Statements Of Resultsmentioning

confidence: 99%

Section: Statements Of Resultsmentioning

confidence: 99%

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Nonlinear anisotropic elliptic equations in with variable exponents and locally integrable data

Mokhtari

2016

Math Methods in App Sciences

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“…The spaces

X_{0}^{1 MathClass-punc, \overset{p}{overrightarrow}} (Ω)

W_{x_{i} MathClass-punc, 0}^{1 MathClass-punc, p_{i}} (Ω)

, and

X MathClass-rel= {double-struckL}^{\overset{p}{overrightarrow}} (0 MathClass-punc, T MathClass-punc; X_{0}^{1 MathClass-punc, \overset{p}{overrightarrow}} (Ω))

were introduced in .…”

Section: Hypotheses and Statement Of Resultsmentioning

confidence: 99%

“…In the isotropic case

p_{1} MathClass-rel= p_{2} MathClass-rel= MathClass-rel\dots MathClass-rel= p_{N} MathClass-rel= \overset{p}{falsemml-overline} MathClass-rel= p

, it has been proved in that there exists a weak solution

u MathClass-rel\in L^{q} (0 MathClass-punc, T MathClass-punc; W_{0}^{1 MathClass-punc, q} (Ω))

to nonlinear parabolic equations with

q MathClass-rel\in [1 MathClass-punc, p MathClass-bin- [N MathClass-bin/ (N MathClass-bin+ 1)])

when μ and μ 0 are two Radon bounded measures and there exists a weak solution

u MathClass-rel\in L^{q} (0 MathClass-punc, T MathClass-punc; W_{0}^{1 MathClass-punc, q} (Ω))

with

q MathClass-rel= p MathClass-bin- [N MathClass-bin/ (N MathClass-bin+ 1)]

when μ 0 ∈ L γ (Ω) and μ ∈ L 1 (0, T ; L γ (Ω)). Recently, for anisotropic parabolic problem with measures as data, I proved in within a different framework of that of the existence of a weak solution