19 de diciembre de 2025
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[VIDEO] Julio César arremete contra Boric: «No puede ir a ver Vóley si tiene la embarrada»

EPCH253 mins

El conductor de CHV criticó públicamente al presidente Gabriel Boric por la difícil situación que están experimentando los niños en la región de Atacama.

Julio César Rodríguez hizo llegar un mensaje contundente al Presidente Gabriel Boric respecto a la alarmante falta de condiciones en las escuelas de la Región de Atacama, situación que ha llevado a los profesores a realizar un prolongado paro.

Durante la transmisión del programa matutino Contigo en la Mañana, Julio César interrumpió la cobertura de los Juegos Panamericanos Santiago 2023 para expresar su indignación: «Si bien los Panamericanos son un evento maravilloso, no podemos permitirnos ser felices mientras continúen ocurriendo estas situaciones en Atacama», manifestó.

«La situación actual de los niños y la educación en Atacama es alarmante», reprochó «JC que le dicen», recordando que los docentes, quienes llevan más de 50 días en huelga, hicieron un llamado al Presidente Boric el martes «para que intervenga» en la difícil situación.

Fue precisamente ese día en el que se observó al mandatario en el partido del equipo chileno de vóley playa femenino. Incluso se le vio jugando junto a la gente.»

Mensaje de Julio César Rodríguez al presidente Boric

Rodríguez no pasó por alto la escena de Boric en las gradas y le reprochó que disfrute de los Juegos mientras Atacama está en una situación difícil.

«Señor Presidente, no puede ir a ver vóleibol si Atacama está en esta situación difícil. Los niños conviven con ratones, apagan la luz con un lápiz Bic porque no tienen interruptor, los baños están en mal estado, el agua está cortada», expresó el presentador de Chilevisión.

«Nadie quiere trabajar para el Estado de Chile porque tienen deudas enormes. Por lo tanto, en Atacama estamos viviendo una situación muy grave con nuestros niños», insistió.

Sobre este tema, el ministro de Educación, Nicolás Cataldo, afirmó que llegarán «hasta las últimas consecuencias en la identificación de responsabilidades, en el uso de los recursos públicos y, por supuesto, en algo tan delicado como poner en riesgo la estabilidad administrativa y financiera de un servicio local como el de Atacama, que está atravesando una situación muy compleja en la actualidad».

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    Below is a short «starter‑pack» bibliography that covers most of the standard results one needs when you try to prove existence (and sometimes uniqueness or regularity) for a weak solution of

    [
    -\Delta u=f(u)\quad\textin \Omega,\qquad u=0 \;\hboxon \partial\Omega,
    ]

    or any other elliptic equation written in the form

    \(Lu=g(x,u,
    abla u)\) with \(L\) self‑adjoint and coercive.

    > Notation – All books are given in a minimal «core» form; you can pick
    the edition that is most convenient for you (hardback, paperback, PDF).
    In particular, all references to Sobolev Spaces
    by Adams & Fournier or similar texts are meant in the sense of the classic Sobolev space theory;
    if you prefer more modern notation (e.g. using \(W^k,p\) instead of \(H^k\)), that is fine – the material remains the same.

    |
    | Reference | Why it matters |

    |—|———–|—————-|
    | 1 | Folland, Real Analysis (chapters on Sobolev spaces and distributions)
    | Gives a clean introduction to Sobolev spaces, weak derivatives,
    and distribution theory. |
    | 2 | Evans, Partial Differential Equations | Classic PDE text that covers elliptic regularity in detail, with clear proofs of Schauder and \(L^p\)
    estimates. |
    | 3 | Gilbarg & Trudinger, Elliptic Partial Differential Equations of Second
    Order | The definitive reference for second‑order elliptic equations; contains the full proof
    of the interior Schauder estimate (Theorem 6.30).
    |
    | 4 | Ladyzhenskaya & Ural’tseva, Linear and Quasi‑Linear Elliptic Equations | Provides \(W^2,
    p\) estimates for uniformly elliptic operators with
    measurable coefficients; useful for understanding the role of \(C^\alpha\) regularity.
    |
    | 5 | Evans, Partial Differential Equations (Chapter 7) |
    Gives a modern exposition of interior regularity for elliptic equations,
    including both Hölder and Sobolev estimates. |

    The book by Gilbarg & Trudinger is an excellent reference for the theory of linear
    uniformly elliptic operators. It contains detailed proofs of Schauder and \(L^p\)
    estimates, as well as examples illustrating the necessity of
    each hypothesis.

    —

    3. Sketches of Proofs

    Below are outlines that capture the main ideas without getting lost in technicalities.
    The details can be found in the cited references.

    3.1. Interior Schauder Estimate (Theorem 2)

    Idea:

    Reduce to a constant‑coefficient equation by freezing coefficients at a point \(x_0\).

    Consider the difference \(w = u – v\), where \(v\) solves
    \(\mathcal L_x_0 v = f(x_0)\) in a ball.

    Show that \(w\) satisfies an elliptic equation with
    small data (because coefficients and RHS vary slowly).

    Use a perturbation argument: the constant‑coefficient problem has known Schauder
    estimates; the smallness allows absorbing terms.

    Key Steps:

    Freezing Coefficients: For any ball \(B_r(x_0)\), define \(\tilde a_ij(x) = a_ij(x_0)\).
    Since \(a_ij\in C^\alpha\), the oscillation of \(a_ij – \tilde a_ij\) is bounded by \(C
    r^\alpha\).

    Comparison Function: Let \(w\) solve

    [
    L_\tilde a w = f, \quad L_\tilde a u = f,
    ]
    with same boundary data on \(\partial B_r(x_0)\).
    Then \(v = u – w\) satisfies
    [
    L_\tilde a v = (L – L_\tilde a)u.
    ]

    Estimate RHS: The right-hand side is bounded by \(C
    r^\alpha |Du|\).

    Apply Interior Schauder Estimate to \(v\): Since the coefficients are constant, we have

    [
    |Dv|_C^0,\gamma(B_r/2) \le C r^\gamma | (L – L_\tilde a)u |_C^0(B_r).
    ]

    Combine with Estimate for \(u\): Use the standard interior estimate for \(u\) in terms of its sup norm
    and RHS.

    Iteration: Iterate this argument on smaller balls, leading to improved Hölder regularity.

    Conclusion: After enough iterations, we obtain that \(
    abla u\) is Hölder continuous with exponent \(\alpha = 1 – \fracnp\), because
    each iteration improves the exponent by a fixed amount determined by \(p\).

    Thus, the final result: For \(f \in L^p(\Omega)\), \(u\)
    solves \(-\Delta u = f\) with Dirichlet boundary conditions.
    Then \(
    abla u \in C^0,\alpha(\overline\Omega)\) where \(\alpha = 1 – n/p\).
    The proof uses a combination of local regularity, the Sobolev embedding theorem, and a bootstrapping argument.

    This explanation is an attempt to illustrate a standard
    result in PDE theory. For a rigorous proof, one should consult
    advanced texts on elliptic partial differential equations or functional analysis.

    We have a conversation: user says «Sure! I’d be happy to walk you through the proof of that regularity theorem.»
    Assistant gave a long explanation. Now user
    says «Sure! I’d be happy to walk you through the proof …».
    Actually the user repeats? They say: «Sure! I’d be happy to walk you through the proof of that regularity theorem.» That is a statement, but
    maybe they want something else: The assistant’s last message was a general explanation. Now the user says something like:
    «Sure! I’d be happy to walk you through the proof of that regularity theorem.» But
    it’s likely they want the assistant to proceed with a step-by-step detailed proof or
    maybe ask for clarification.

    Given the conversation, I think the user might want to go into deeper detail or confirm steps.
    Perhaps they want to ask: «Explain how to apply De Giorgi’s lemma» or «Show how Morrey’s inequality yields Hölder continuity».
    Or they may be asking to start from the beginning: «Prove that a weak solution of Laplace equation is smooth».

    Alternatively, maybe the user wants the assistant to give a
    full rigorous proof. The last message: «Prove that a weak solution u to -Δu=0 is locally C^∞.» The assistant gave an outline.
    Now the user says «Prove» and then gives a bullet list of
    steps: 1) Show L2 estimate; 2) Use Sobolev embedding etc.

    Wait, reading carefully: The last message from the user:

    Prove that a weak solution u to -Δu=0 is locally C∞.

    – Show that u satisfies an L^2 estimate
    on each ball
    B_R ⊂ Ω:
    ∫_B_R |∇u|^2 dx ≤ C R^-2 ∫_B_2R |u|^2 dx
    – Use Sobolev embedding to deduce that u ∈ H^1(B_R) and then in L^p for
    some p>2.

    But earlier the system says: «Prove that a weak solution u to -Δu=0 is locally C∞.» And we are given the steps.

    Thus the question likely asks us to show that a harmonic function (a
    weak solution of Laplace’s equation) has interior regularity, specifically local boundedness and smoothness.
    The standard approach: use Caccioppoli inequality (energy estimate), then Moser iteration or De Giorgi technique to get L∞ bounds, then bootstrap using elliptic
    PDE theory.

    But the question as phrased may be simpler: «Prove that a weak solution u to -Δu=0 is locally bounded» perhaps.
    Actually, the given steps are about proving local boundedness of harmonic functions from
    energy estimates and Sobolev inequalities. The final aim might be
    to show that any weak solution to Laplace’s equation is smooth.

    Given the structure: first Caccioppoli inequality gives control over ∇u in L^2 by u
    itself; then Sobolev embedding gives a reverse Hölder inequality for u, i.e., an integral estimate that
    yields local boundedness. Then using Moser iteration or De
    Giorgi–Nash–Moser theory, we get continuity and Harnack
    inequality.

    Thus the answer likely: «Using these estimates we can show that any weak solution of the Laplace equation is locally bounded (in fact Hölder continuous), and thus by elliptic regularity it is smooth.»

    So I will write: The estimates imply a reverse Hölder inequality for u, which yields local boundedness.
    Then using standard arguments (Moser iteration or De Giorgi’s
    method) we obtain that u is locally Hölder continuous.
    By bootstrapping the PDE we get higher regularity: all derivatives of
    u are smooth inside Ω. In other words, any weak solution of Δu=0 in a domain is
    C∞. This is the classical elliptic regularity result
    for harmonic functions.

    Thus, the estimates prove that the solutions to this linear elliptic equation are
    actually analytic (real-analytic). So the PDE has smooth solutions
    given weak initial data.

    Hence: The estimate shows that u is locally bounded by its L^2 norm dianabol and testosterone cycle thus can be improved
    to local Hölder continuity; subsequently, standard elliptic regularity yields C∞.
    Thus the conclusion.The inequality tells us that
    a solution \(u\) of a linear elliptic equation
    (\(Lu=f\), say) is not only locally in \(L^2\) but also has its supremum
    controlled by an \(L^2\)-norm over a slightly larger set.
    This estimate
    is the starting point for all of the usual regularity results.

    —

    1. From the inequality to local boundedness

    The right–hand side is a multiple of the average of \(|u|^2\) on the
    larger ball \(B_R+3\varepsilon(x)\).

    Since an \(L^2\)-average controls the essential supremum
    of a function,
    the inequality shows that for every point \(x\) and each small
    \(0<\varepsilon<

    [
    |u(x)| \le C\,R^-N/2\left(\int_B_R+3\varepsilon(x)|u(y)|^2dy\right)^1/2.
    ]

    Thus \(u\) is locally bounded.

    In particular, the supremum over a compact set is finite; the
    \(L^\infty\)-norm of \(u\) on any relatively compact domain can be
    controlled by its \(L^2\)-mass in a slightly larger neighbourhood.
    This estimate is a key step toward proving that \(u\) belongs to
    \(W^1,2\), and it shows that an \(L^2\) function satisfying the
    Poisson equation with integrable data cannot be too large locally.
    The argument uses only elementary inequalities (Hölder’s inequality,
    Poincaré inequality) together with the properties of mollifiers to
    regularise the problem.

    —

    So, the estimate tells us that the \(L^2\)-norm of a solution to the Poisson equation is bounded by the sum of two terms: the norm of its gradient and the norm of the function it satisfies. This helps in showing regularity or convergence properties for solutions in Sobolev spaces. If you want a deeper understanding, we can look into the proofs, but this overview should give you a clear sense of what the estimate achieves.

    Responder

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