Regularity theory for non-autonomous problems with a priori assumptions

We study weak solutions and minimizers $u$ of the non-autonomous problems $\operatorname{div} A(x, Du)=0$ and $\min_v \int_\Omega F(x,Dv)\,dx$ with quasi-isotropic $(p, q)$-growth. We consider the case that $u$ is bounded, H\"older continuous or lies in a Lebesgue space and establish a sharp connection between assumptions on $A$ or $F$ and the corresponding norm of $u$. We prove a Sobolev--Poincar\'e inequality, higher integrability and the H\"older continuity of $u$ and $Du$. Our proofs are optimized and streamlined versions of earlier research that can more readily be further extended to other settings. Connections between assumptions on $A$ or $F$ and assumptions on $u$ are known for the double phase energy $F(x, \xi)=|\xi|^p + a(x)|\xi|^q$. We obtain slightly better results even in this special case. Furthermore, we also cover perturbed variable exponent, Orlicz variable exponent, degenerate double phase, Orlicz double phase, triple phase, double variable exponent as well as variable exponent double phase energies and the results are new in most of these special cases

Maximal regularity for local minimizers of non-autonomous functionals

We establish local C-1;alpha-regularity for some alpha is an element of (0, 1) and C-alpha-regularity for any alpha is an element of (0, 1) of local minimizers of the functionalnu bar right arrow integral(Omega)phi(x, vertical bar D nu vertical bar)dx,where phi satisfies a(p, q)-growth condition. Establishing such a regularity theory with sharp, general conditions has been an open problem since the 1980s. In contrast to previous results, we formulate the continuity requirement on phi in terms of a single condition for the map(x, t) bar right arrow phi(x, t), rather than separately in the x- and t -directions. Thus we can obtain regularity results for functionals without assuming that the gap q=p between the upper and lower growth bounds is close to 1. Moreover, for phi(x, t) with particular structure, including p-, Orlicz-, p(x)- and double phasegrowth, our single condition implies known, essentially optimal, regularity conditions. Hence, we handle regularity theory for the above functional in a universal way

Regularity Theory for Non-autonomous Partial Differential Equations Without Uhlenbeck Structure

We establish maximal local regularity results of weak solutions or local minimizers of div A(x, Du) = 0 and min(u) integral(Omega) F(x, Du)dx,providing new ellipticity and continuity assumptions on A or F with general (p, q)-growth. Optimal regularity theory for the above non-autonomous problems is a long-standing issue; the classical approach by Giaquinta and Giusti involves assuming that the nonlinearity F satisfies a structure condition. This means that the growth and ellipticity conditions depend on a given special function, such as t(p), phi (t), t(p(x)), t(p) +a(x)t(q), and not only F but also the given function is assumed to satisfy suitable continuity conditions. Hence these regularity conditions depend on given special functions. In this paper we study the problem without recourse to, special function structure and without assuming Uhlenbeck structure. We introduce a new ellipticity condition using A or F only, which entails that the function is quasi-isotropic, i.e. it may depend on the direction, but only up to a multiplicative constant. Moreover, we formulate the continuity condition on A or F without specific structure and without direct restriction on the ratio q/p of the parameters from the (p, q)-growth condition. We establish local C-1,C-alpha-regularity for some alpha is an element of (0, 1) and C-alpha-regularity for any alpha is an element of (0, 1) of weak solutions and local minimizers. Previously known, essentially optimal, regularity results are included as special cases.</p

Adsorptive and kinetic characterization of aqueous zinc removal by biochars

Biochars have shown a great potential to treat stormwater runoff contaminated with heavy metals due to their favorable physical and chemical characteristics. Biochar materials were produced from pyrolysis of oak tree and wood at 400C and 450C respectively, and their Zn adsorption behavior from aqueous solutions were evaluated to assess their applicability as a filter media for stormwater treatment. Two adsorption isotherm models, Freundlich and Langmuir, were used to fit the batch-scale experimental data. The kinetics of Zn adsorption was investigated under two contrasting physical condition (stagnant vs. agitated). The adsorption isotherm was better fitted with the Langmuir model (R2 = 0.99) than the Freundlich model (R2 = 0.62-0.72). Oak tree biochar (~ 21,400 mg kg-1) outperformed wood biochar (~ 6,100 mg kg-1) in the Zn adsorption due to higher molar ratio of oxygen to carbon in the oak tree biochar. The Zn adsorption by the biochars were less effective under stagnant condition, suggesting that external energy for agitation is needed when considering biochar as a stormwater filter media. Overall the kinetics data of Zn adsorption fitted well with the pseudo-second order model (R2 = 0.99), indicating that chemisorption was dominant mechanism for the Zn adsorption onto the biochars. This study highlights a potential for biochar to be an effective adsorbent to remove Zn with relatively short contact time for stormwater and industrial applications

Antitumor effect of TW-37, a BH3 mimetic in human oral cancer

TW-37 is a small molecule B cell lymphoma-2 (Bcl-2) homology 3 mimetic with potential anticancer activities. However, the in vivo anti-cancer effect of TW-37 in human oral cancer has not been properly studied yet. Here, we attempted to confirm antitumor activity of TW37 in human oral cancer. TW-37 significantly inhibited cell proliferation and increased the number of dead cells in MC-3 and HSC-3 human oral cancer cell lines. TW-37 enhanced apoptosis of both cell lines evidenced by annexin V/propidium iodide double staining, sub-G1 population analysis and the detection of cleaved poly (ADP-ribose) polymerase and caspase-3. In addition, TW-37 markedly downregulated the expression of Bcl-2 protein, while not affecting Bcl-xL or myeloid cell leukemia-1. In vivo, TW-37 inhibited tumor growth in a nude mice xenograft model without any significant liver and kidney toxicities. Collectively, these data reveal that TW-37 may be a promising small molecule to inhibit human oral cancer.This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science ICT & Future Planning [2019R1A2C1085896]

Boundary partial Holder regularity for elliptic systems with non-standard growth

We investigate regular points on the boundaries of elliptic systems with non-standard growth, in particular, so-called Orlicz growth. A regular point on the boundary in this paper is a point for which a weak solution to a system is Holder continuous in a neighborhood. Here, we assume that the boundary of a domain and the boundary data are C1, and that a system has VMO (vanishing mean oscillation) type coefficients

Partial continuity for a class of elliptic systems with non-standard growth

We study partial Holder continuity of weak solutions to elliptic systems with variable non-standard growth, which are related to the function $\Phi(x,t):=t^{p(x)}\log(e+t)$. We prove that weak solutions are Holder continuous for any Holder exponent, except Lebesgue measure zero sets, if systems satisfy certain continuity assumptions. In particular, the variable exponent functions p(.) are assumed to satisfy so-called vanishing log-Holder continuity

Calderón–Zygmund estimates in generalized Orlicz spaces

Abstract We establish the $W^{2,\varphi(⋅)}$-solvability of the linear elliptic equations in non-divergence form under a suitable, essentially minimal, condition of the generalized Orlicz function $\varphi(⋅)=\varphi(x,t)$, by deriving Calderón–Zygmund type estimates. The class of generalized Orlicz spaces we consider here contains as special cases classical Lebesgue and Orlicz spaces, as well as non-standard growth cases like variable exponent and double phase growth