Gradient Hölder regularity for parabolic normalized p(x,t)-Laplace equation

Acetogenins from the Stem of Uvaria rufa and Their Cytotoxic Activity

“…6.99). 24,25 The placement of the hydroxy group at C-6 was further confirmed by a TOCSY correlation between H-3 (δ H 2.30) and H-6 (δ H 3.62), indicating the proximity of the OH group to the lactone part. Further corroboration came from the peak in the MALDI-MS/MS at m/z 175 corresponding to a lithiated A 1 ion, whereas in C-5-hydroxylated narumicins this ion is showing at m/z 161 (Figure 1).…”

“…21−23 The signal of one proton at δ 1 and 2). 23,24 A third OH appeared in the 1 H NMR spectrum at δ H 3.62 (H-6) and in the 13 C NMR spectrum at δ C 71.7 (C-6). Its position at C-6 was predicted based on the chemical shifts at H-3 (δ H 2.30) and H-35 (δ H 7.02), which are more shielded than with C-5 hydroxylated (δ H-3 ca.…”

“…The signals at δ H 3.39 integrating for two protons (H-15, -24) and between δ H 3.80−3.90 integrating for four protons (H-16, -19, -20, -23) and those at δ C 83.4, 83.3 (C-16, 23), 81.9 (C-19), 81.9 (C-20), and 74.3 and 74.2 (C-15, 24) in the 1 H and 13 C NMR spectra indicated the presence of an adjacent bis-THF ring system flanked by two OH groups, with a threo/trans/threo/trans/threo relative configuration (Tables 1 and 2). 23,24 This relative configuration can result in pseudo-symmetry across the THF pattern, and only two signals between δ C 81 and δ C 84 ppm and one signal around δ C 74 ppm can sometimes be observed, 23 as in the case of narumicin I (5) (Table S2). However, the presence of two additional OH groups on one side of the OH-flanked bis-THF ring unit disturbs the symmetry.…”

A second-order Sobolev regularity for p(x)-Laplace equations

“…It arises naturally from a two-player zero-sum stochastic differential game (SDG) with probabilities depending on space and time, please see [29]. Parviainen-Ruosteenoja, in [29] proved the Hölder and Harnack estimates for a more general game that was called p(x, t)-game without using the PDE techniques and showed that the value functions of the game converge to the unique viscosity solution of the Dirichlet problem to the normalized p(x, t)-parabolic equation (n + p(x, t))u t (x, t) = ∆ N p(x) u(x, t) Moreover, Chao estabalished the interior Hölder regularity of the spatial gradient of viscosity solutions in [13].…”

“…To prove Theorem 1.1, given any fixed smooth domain U ⊂⊂ Ω, and for ǫ ∈ (0, 1], let u ǫ be a viscosity solution to the regularized equation (1.4), we know that u ǫ has the C 1,α loc -regularity in the spatial variable and the C 1, 1+α 2 -regularity in the time variable, especially Du ǫ ∈ L ∞ (U T ) uniformly in ǫ > 0; see [13].…”