Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems

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Let ๐‘‹ be a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual space ๐‘‹โˆ—. Let ๐‘‡: ๐‘‹ โŠ‡ ๐ท(๐‘‡) โ†’ 2๐‘‹โˆ— be maximal monotone and ๐‘† : ๐‘‹ โŠ‡ ๐ท(๐‘†) โ†’ ๐‘‹โˆ— quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space ๐‘Š โŠ‚ ๐ท(๐‘†), dense and continuously embedded in ๐‘‹. Assume, further, that there exists ๐‘‘ โ‰ฅ 0 such that โŸจ๐˜ทโˆ— + ๐‘†๐‘ฅ, ๐‘ฅโŸฉ โ‰ฅ โˆ’dโ€–๐‘ฅโ€–ยฒ for all ๐‘ฅ โˆˆ ๐ท(๐‘‡) โˆฉ๐ท(๐‘†) and ๐˜ทโˆ— โˆˆ ๐‘‡๐‘ฅ. New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the type ๐‘‡+๐‘†. A partial positive answer for Nirenbergโ€™s problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operator ๐ฟ : ๐‘‹ โŠ‡ ๐ท(๐ฟ) โ†’ ๐‘‹ โˆ— is given as a result of surjectivity of ๐ฟ + ๐‘†, where ๐‘† is of type (๐‘€) with respect to ๐ฟ.These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the last section, an example is provided addressing existence of weak solution in ๐‘‹ = ๐ฟ๐‘(0, ๐‘‡;๐‘Šโ‚€ยน,๐‘ (ฮฉ)) of a nonlinear parabolic problem of the type ๐‘ข๐‘กโˆ’ ฮฃ๐‘›๐‘–=1(๐œ•/๐œ•๐‘ฅ๐‘–)๐‘Ž๐‘– (๐‘ฅ, ๐‘ก, ๐‘ข, โˆ‡๐‘ข) = ๐‘“(๐‘ฅ, ๐‘ก), (๐‘ฅ, ๐‘ก) โˆˆ ๐‘„; ๐‘ข(๐‘ฅ, ๐‘ก) = 0, (๐‘ฅ, ๐‘ก) โˆˆ ๐œ•ฮฉ ร— (0, ๐‘‡); ๐‘ข(๐‘ฅ, 0) = 0, ๐‘ฅ โˆˆ ฮฉ, where ๐‘ > 1, ฮฉ is a nonempty, bounded, and open subset of R๐‘, ๐‘Ž๐‘–: ฮฉ ร— (0,๐‘‡) ร— โ„ ร— โ„๐‘ โ†’ โ„ (๐‘– = 1, 2, . . . , ๐‘›) satisfies certain growth conditions, and ๐‘“ โˆˆ ๐ฟ๐‘' (๐‘„), ๐‘„ = ฮฉ ร— (0,๐‘‡), and ๐‘' is the conjugate exponent of ๐‘.




Teffera M. Asfaw, "Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems", Abstract and Applied Analysis, vol. 2015, Article ID 357934, 11 pages, 2015.