We establish universal interior H\"older and fractional energy estimates for stable solutions of $(-\Delta)^su=f(u)$, $0<s<1$, with $f\in C^1_{\mathrm{loc}}(\mathbb{R})$ nonnegative and convex. The bounds depend only on a weighted $L^1$ norm of the solution and hold under optimal conditions on the dimension and the fractional order. In particular, the half-Laplacian estimate holds through dimension eight. The estimates extend to stable finite-energy Dirichlet $L^1$-weak solutions without an a priori local boundedness assumption, and an explicit logarithmic solution proves optimality in this class. Applications include global bounds in convex domains, bounded extremal solutions, and a Liouville theorem.
We investigate interior regularity for stable solutions of the semilinear elliptic equation $-\Delta u=f(u)$ in an open set $\Omega\subseteq \mathbb R^n$, where $f$ is locally Lipschitz and may change sign. We establish sharp interior H\"older estimates for stable classical solutions in every dimension $n\le 9$, thereb...
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
This paper establishes the exponential $L^2$-stability of the unique solutions to initial-boundary value problems for linear parabolic partial differential equations with general drift and zero-order coefficients in bounded domains. The key idea lies in constructing a suitable Dirichlet form with respect to a weighted...
We establish an interior $C^2$ estimate for locally bounded plurisubharmonic Bedford-Taylor solutions of the complex Monge--Amp\`ere equation $\det(u_{i\bar j})=f$ under weak quadratic growth bounds on sufficiently large balls, where $f$ is positive and Dini-continuous. We assume a distributional lower bound for the La...
We study local and global properties of positive solutions to the equation $-\Delta u=u^p+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p,q$ are parameters and $M>0$. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate aux...
We develop a unified approach to Schauder estimates for scalar parabolic equations \[ \partial_t u-\operatorname{div}\big(\mathcal{A}(x,t,Du)\big)=f(x,t) \] with general Orlicz growth and coefficients that are H\"older continuous in the spatial variables and merely measurable in time. Combining intrinsic gradient estim...
Carlo Alberto Antonini· 0 citations
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