We develop a fixed-reference weighted $L^2$ theory for terminal-value fully nonlinear parabolic equations. A prescribed uniformly elliptic diffusion starting from a point determines both the linear reference operator and a space--time occupation measure; time weights of order $\alpha$ accommodate the point-start singularity. For the associated linear backward equation, we identify the optimal source-to-intrinsic-Hessian norm $C_w(X)$ and derive bounds on it from a moving-measure Bochner identity. We compute the sharp Brownian benchmark $C_\alpha^{\mathrm{Br}}=\sup_{n\in\mathbb{N}, n\geq 2}2\sqrt{n(n-1)}/(n-1+\alpha)$, prove that it is a universal tangent lower bound, and show that it is attained by regular weights for time-inhomogeneous affine Gaussian diffusions and is the infimum over such weights for normalized state-dependent covariances. Under the standing data and lower-order assumptions, the Hessian estimate yields our main nonlinear result: if the driver's Hessian Lipschitz constant satisfies $C_w(X)L_H<1$, then the equation has a unique weighted Sobolev solution and the Picard iteration on the source converges geometrically, with no smallness condition on the value or gradient channels. For the Brownian model at order $1/2$, counterexamples in dimensions $d\geq7$ show that this strict threshold cannot be uniformly improved within the occupation-Sobolev class. For weights of order below one, additional spatial regularity and unweighted integrability of the data yield an unweighted spatial jet and a unique solution in the regular fixed-reference Markovian class for second-order backward stochastic differential equations (2BSDEs) defined here.
We study the existence and multiplicity of sign-changing solutions of the semilinear elliptic equation
$$ -\Delta _g u + u = f(u) \quad \text {on } \mathbb {S}^2, $$
where
$$(\mathbb {S}^2,g)$$
denotes the two-dimensional unit sphere endowed with a smooth Riemannian metric,
$$\Delta _g$$
is the Laplace–Beltrami operator, and
f
may exhibit critical exponential growth in the sense of the Trudinger-Moser inequality. Our method combines a variational construction on geodesic lunes with a global reflection scheme. We first solve a Dirichlet problem on a fixed lune and show that the corresponding mountain-pass level remains strictly below the Trudinger-Moser threshold. Reflecting this solution across meridians and alternating its sign yields a sequence of global solutions with prescribed nodal patterns and quantized energy levels when
f
is odd. For small nonsymmetric perturbations of the nonlinearity, we employ a Lyapunov-Schmidt reduction to adjust the resulting multi-lune configurations, thereby proving the persistence of these sign-changing solutions.
Unknown authors· Annali di Matematica Pura ed...· 0 citations
In this paper, let $u\in C^4(B_{10})$ with $D^2u\geq -KI$ define a $2$-admissible graph $M=\{(x,u(x)):x\in B_{10}\}\subset\R^{n+1}$ satisfying \[ \sigma_2(\kappa[u])=f(x). \] We prove an interior curvature estimate depending on the Lipschitz norm of the right-hand sides. The proof combines a shifted Jacobi inequality for \(b=\log(H+J_0)\), a parallel hypersurface transformation that makes the Newton tensor uniformly elliptic and local boundedness estimate then reduces the pointwise bound to a weighted $L^1$ estimate, which is completed using the Jacobi energy inequality and integration by parts.
Let $n\ge2$ and let $u$ be a smooth 2-convex and semi-convex solution of \[ \frac{\sigma _2(D^2u)}{\sigma _1(D^2u)}=f(x). \] We prove an interior Hessian estimate depending on the Lipschitz norm of $f$. The proof combines the integral approach of Chen--Jian--Zhou with the algebraic reduction of the quotient equation to a $\sigma _2$ structure. The main new point is a shifted algebraic inequality that yields a shifted trace Jacobi inequality in divergence form for $\log(\Delta u+a)$. We work with the linearized operator $G=(\Delta u-f)I-D^2u$ of the equivalent equation $\sigma_2(D^2u)=f \Delta u$. The almost divergence-free identity \(\partial_iG_{ij}=-f_j\) enables us to control the \(\Delta f\) term by integration by parts solely in terms of the Lipschitz norm of \(f\). A Legendre--Lewy transformation converts the resulting degenerate divergence-form equation into a uniformly elliptic one. The estimate then follows from a mean-value inequality together with a weighted energy argument. As an application, in dimension two we obtain interior $C^2$ regularity for convex viscosity solutions with positive Lipschitz right-hand side. Moreover, our counterexamples show that the Lipschitz regularity required of the right-hand side is optimal.
We establish the interior $C^{1,α}$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t = |Du|^γF(D^2u) + f.$$ For this purpose, we prove the well-posedness of the regularized Dirichlet problem \begin{equation*} \left\{ \begin{aligned} u_t&=(1+|Du|^2)^{γ/2}F(D^2u) &&\text{in $Q_1$} \newline u&=φ&&\text{on $\partial_p Q_1$}. \end{aligned}\right. \end{equation*} Our approach utilizes the Bernstein method with approximations in view of difference quotient.
We establish scale-invariant interior $C^{1,\alpha}$ estimates for bounded viscosity solutions of $(-\Delta)^su+b\cdot\nabla u=f$ for $s\in[1/2,1)$ with locally H\"older $b$ and $f$. The critical case uses Silvestre's parabolic theorem; the subcritical case uses Schauder estimates and interpolation. Applying this viscosity estimate to drifted Green sections, for finite Radon data above the critical order we obtain sharp solution and gradient potentials of orders $2s$ and $2s-1$, together with weak-*--to--strong local $W^{1,1}$ stability; hence the Green-potential SOLA is approximation-independent. For the normalized whole-space kernels, we identify the classical second-order limits as $s\uparrow1$, including the logarithmic kernel in dimension two; at the critical order, the whole-space gradient becomes a zero-order singular integral. For zero-exterior problems with compactly supported drift, we also prove $u/d^s\in C^{s-\varepsilon}(\overline\Omega)$ and identify the obstruction when the drift reaches the boundary.
Let $n\ge2$, $1p-1$. We prove that every globally bounded fractional $p$-harmonic function is locally $C^{1,\alpha}$ for some $\alpha=\alpha(n,p,s)>0$. This settles the open problem of interior gradient H\"older regularity in the singular range throughout the natural first-order regime $sp>p-1$. The proof combines an affine-invariant improvement-of-flatness argument with a Liouville theorem for globally Lipschitz entire solutions. In the large-slope regime, the shifted Bregman energies converge to an anisotropic stable form of order $sp-p+2>1$. In the bounded-slope regime, the Liouville theorem follows from rigidity of extremal secants, a recurrent blow-down argument, and a directional Morrey-Kato estimate for the singular linearized kernel. An affine Campanato argument controls the variation of the best affine approximations across scales. These estimates yield a scale-invariant decay of the affine excess and hence the local $C^{1,\alpha}$ estimate.
Chao Zhang· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
