25,143 research outputs found
Existence and multiplicity of solutions to equations of Laplacian type with critical exponential growth in
In this paper, we deal with the existence and multiplicity of solutions to
the nonuniformly elliptic equation of the N-Lapalcian type with a potential and
a nonlinear term of critical exponential growth and satisfying the
Ambrosetti-Rabinowitz condition. In spite of a possible failure of the
Palais-Smale compactness condition, in this article we apply minimax method to
obtain the weak solution to such an equation. In particular, in the case of
Laplacian, using the minimization and the Ekeland variational principle, we
obtain multiplicity of weak solutions.
Finally, we will prove the above results when our nonlinearity doesn't
satisfy the well-known Ambrosetti-Rabinowitz condition and thus derive the
existence and multiplicity of solutions for a much wider class of nonlinear
terms .Comment: 30 pages. First draft in November, 201
Degenerate complex Hessian equations on compact K\"ahler manifolds
Let be a compact K\"ahler manifold of dimension and fix
such that . We prove that any -sh
function can be approximated from above by smooth -sh functions. A
potential theory for the complex Hessian equation is also developed which
generalizes the classical pluripotential theory on compact K\"ahler manifolds.
We then use novel variational tools due to Berman, Boucksom, Guedj and Zeriahi
to study degenerate complex Hessian equations
Adaptive Nonparametric Image Parsing
In this paper, we present an adaptive nonparametric solution to the image
parsing task, namely annotating each image pixel with its corresponding
category label. For a given test image, first, a locality-aware retrieval set
is extracted from the training data based on super-pixel matching similarities,
which are augmented with feature extraction for better differentiation of local
super-pixels. Then, the category of each super-pixel is initialized by the
majority vote of the -nearest-neighbor super-pixels in the retrieval set.
Instead of fixing as in traditional non-parametric approaches, here we
propose a novel adaptive nonparametric approach which determines the
sample-specific k for each test image. In particular, is adaptively set to
be the number of the fewest nearest super-pixels which the images in the
retrieval set can use to get the best category prediction. Finally, the initial
super-pixel labels are further refined by contextual smoothing. Extensive
experiments on challenging datasets demonstrate the superiority of the new
solution over other state-of-the-art nonparametric solutions.Comment: 11 page
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