Aksioma aksioma

Sobolev spaces[edit]
Sobolev spaces, denoted by H^s or W^{s, 2}, are Hilbert spaces. These are a special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of partial differential equations.^[26] They also form the basis of the theory of direct methods in the calculus of variations.^[27]
Uchun s manfiy bo'lmagan butun son va A. S. R ^N, Sobolev fazosi h ^S(A. S.) o'z ichiga oladi l ² tartibning kuchsiz hosilalari s gacha bo'lgan funksiyalar ham l ². H ^{s ning ichki mahsuloti} (Bac) bu
{\displaystyle \langle f,g\rangle =\int _{\Omega }f(x){\bar {g}}(x)\,\mathrm {d} x+\int _{\Omega }Df(x)\cdot D{\bar {g}}(x)\,\mathrm {d} x+\cdots +\int _{\Omega }D^{s}f(x)\cdot D^{s}{\bar {g}}(x)\,\mathrm {d} x}

bu erda nuqta nuqta hosilasi ichida Evklid fazosi har bir tartibning qisman hosilalari. Sobolev bo'shliqlarini qachon ham aniqlash mumkin s butun son emas.
Sobolev bo'shliqlari spektral nazariya nuqtai nazaridan ham o'rganiladi, aniqrog'i Hilbert kosmik tuzilishiga tayanadi. Agar kupchilik mos domen bo'lsa, unda Sobolev maydonini aniqlash mumkin H ^S(kupchilik) Bessel potentsiallari maydoni sifatida;^[28] taxminan,
{\displaystyle H^{s}(\Omega )=\left.\left\{(1-\Delta )^{-{\frac {s}{2}}}f\;\right|\,f\in L^{2}(\Omega )\right\}\,.}

Bu erda Kuplaciya va (1 − Xnumx−Xnumx-Xnumx-Xnumx-Xnumx) ^-2/s spektral xaritalash teoremasi jihatidan tushuniladi. Sobolev bo'shliqlarining butun son bo'lmagan ta'rifini berishdan tashqari s, ushbu ta'rif ostida ayniqsa kerakli xususiyatlarga ega Furye konvertatsiyasi bu uni o'rganish uchun ideal qiladi psevdodifferensial operatorlar. Ushbu usullardan ixcham Riemann manifoldida foydalanib, masalan, Xodj nazariyasining asosi bo'lgan Xodj dekompozitsiyasini olish mumkin.^[29]