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48
result(s) for
"Goldstein, Paweł"
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Smooth approximation of mappings with rank of the derivative at most 1
It was conjectured that if
f
∈
C
1
(
R
n
,
R
n
)
satisfies
rank
D
f
≤
m
<
n
everywhere in
R
n
, then
f
can be uniformly approximated by
C
∞
-mappings
g
satisfying
rank
D
g
≤
m
everywhere. While in general, there are counterexamples to this conjecture, we prove that the answer is in the positive when
m
=
1
. More precisely, if
m
=
1
, our result yields an almost-uniform approximation of locally Lipschitz mappings
f
:
Ω
→
R
n
, satisfying
rank
D
f
≤
1
a.e., by
C
∞
-mappings
g
with
rank
D
g
≤
1
, provided
Ω
⊂
R
n
is simply connected. The construction of the approximation employs techniques of analysis on metric spaces, including the theory of metric trees (
R
-trees).
Journal Article
Jacobians of W1,p homeomorphisms, case p=n/2
We investigate a known problem whether a Sobolev homeomorphism between domains in
R
n
can change sign of the Jacobian. The only case that remains open is when
f
∈
W
1
,
[
n
/
2
]
,
n
≥
4
. We prove that if
n
≥
4
, and a sense-preserving homeomorphism
f
satisfies
f
∈
W
1
,
[
n
/
2
]
,
f
-
1
∈
W
1
,
n
-
[
n
/
2
]
-
1
and either
f
is Hölder continuous on almost all spheres of dimension [
n
/ 2], or
f
-
1
is Hölder continuous on almost all spheres of dimensions
n
-
[
n
/
2
]
-
1
, then the Jacobian of
f
is non-negative,
J
f
≥
0
, almost everywhere. This result is a consequence of a more general result proved in the paper. Here [
x
] stands for the greatest integer less than or equal to
x
.
Journal Article
Jacobians of W 1 , p homeomorphisms, case p = n / 2
We investigate a known problem whether a Sobolev homeomorphism between domains in Rn can change sign of the Jacobian. The only case that remains open is when f∈W1,[n/2], n≥4. We prove that if n≥4, and a sense-preserving homeomorphism f satisfies f∈W1,[n/2], f-1∈W1,n-[n/2]-1 and either f is Hölder continuous on almost all spheres of dimension [n / 2], or f-1 is Hölder continuous on almost all spheres of dimensions n-[n/2]-1, then the Jacobian of f is non-negative, Jf≥0, almost everywhere. This result is a consequence of a more general result proved in the paper. Here [x] stands for the greatest integer less than or equal to x.
Journal Article
(\\mathbf{C^2}\\)-Lusin approximation of convex functions: one variable case
We prove that if \\(f:(a,b)\\to\\mathbb{R}\\) is convex, then for any \\(\\varepsilon>0\\) there is a convex function \\(g\\in C^2(a,b)\\) such that \\(|\\{f\\neq g\\}|<\\varepsilon\\) and \\(\\Vert f-g\\Vert_\\infty<\\varepsilon\\).
Paper
Smooth approximation of mappings with rank of the derivative at most \\(1\\)
It was conjectured that if \\(f\\in C^1(\\mathbb{R}^n,\\mathbb{R}^n)\\) satisfies \\(\\operatorname{rank} Df\\leq m
Paper
Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms
Cerf and Palais independently proved a remarkable result about extending diffeomorphisms defined on smooth balls in a manifold to global diffeomorphisms of the manifold onto itself. We explain Palais' argument and show how to extend it to the class of homeomorphisms and bi-Lipschitz homeomorphisms. While Palais' argument is surprising, it is elementary and short. However, its extension to bi-Lipschitz homeomorphisms and homeomorphisms requires deep results: the stable homeomorphism and the annulus theorems.
Paper
Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms
In the paper, we address the following problem, often encountered in geometric constructions: having defined an orientation preserving diffeomorphism in a number of `patches' on a manifold \\(M\\), can we extend it to a diffeomorphism of the whole manifold \\(M\\)? We give a positive answer, provided the `patches' are sufficiently regular. Then, we extend the same result to the case of bi-Lipschitz homeomorphisms and homeomorphisms.
Paper
Constructing diffeomorphisms and homeomorphisms with prescribed derivative
We prove that for any measurable mapping \\(T\\) into the space of matrices with positive determinant, there is a diffeomorphism whose derivative equals \\(T\\) outside a set of measure less than \\(\\). We use this fact to prove that for any measurable mapping \\(T\\) into the space of matrices with non-zero determinant (with no sign restriction), there is an almost everywhere approximately differentiable homeomorphism whose derivative equals \\(T\\) almost everywhere.
Paper
Jacobians of \\(W^{1,p}\\) homeomorphisms, case \\(p=n/2\\)
We investigate a known problem whether a Sobolev homeomorphism between domains in \\(\\mathbb{R}^n\\) can change sign of the Jacobian. The only case that remains open is when \\(f\\in W^{1,[n/2]}\\), \\(n\\geq 4\\). We prove that if \\(n\\geq 4\\), and a sense-preserving homeomorphism \\(f\\) satisfies \\(f\\in W^{1,[n/2]}\\), \\(f^{-1}\\in W^{1,n-[n/2]-1}\\) and either \\(f\\) is H\"older continuous on almost all spheres of dimension \\([n/2]\\), or \\(f^{-1}\\) is H\"older continuous on almost all spheres of dimensions \\(n-[n/2]-1\\), then the Jacobian of \\(f\\) is non-negative, \\(J_f\\geq 0\\), almost everywhere. This result is a consequence of a more general result proved in the paper. Here \\([x]\\) stands for the greatest integer less than or equal to \\(x\\).
Paper